The slice Burnside ring and the section Burnside ring of a finite group

The slice Burn side ring and the sectio n Burnside ring of a finite group Serge Bouc Abstract : This pap er introduces tw o new Burnside rings for a finite group G , called the slic e Burnside ring and the se ction Burnside ring . They are built as Grothendieck rings of the ca tegory o f mo rphisms o f G -sets, and of Galois morphisms of G -sets, resp ectively . The well known results on the usua l Burnside ring, co ncerning ghost maps, primitive idempo tent s, and description of the prime sp ectrum, are extended to these rings. It is also shown tha t these tw o r ing s hav e a natural structure o f Green biset functor. The functorial structure of unit groups of these rings is also discussed. AMS Sub ject Classification : 19A22 , 1 8F30. Keyw ords : Burnside ring, Green biset functor, slice, section, unit gr oup. 1. In tro duction This pap er in tro duces tw o v ariations on Burnside rings o f finite gro ups, called the slic e Burnside ring and the se ction Burnside rin g . Both of them are built as Grothendiec k rings of some category of morphis ms of finite G -sets, instead of t he category o f finite G -sets used to build the usual Burns ide ring. The difference b et w een these tw o new Burnside rings is t ha t the slice Burnside ring is built from arbitrary morphisms o f finite G -sets, whereas the section Burnside ring uses only Galois morphis m s of finite G -sets. It turns out that most o f the w ell kno wn prop ertie s of the Burnside ring (see e.g.[3]) extend to the slice Burnside ring and to the section Burnside ring : b oth are commutativ e rings, whic h are fr ee o f finite rank as Z -module. There is an analog ue of Burnside’s theorem : b oth of t hese rings embed in a pro duct of copies of the inte gers, via a gh ost ma p , and this map has a finite cok ernel. After tensoring with Q , b oth rings b ecome split semisimple Q -algebras, and explicit form ulae fo r their primitiv e idemp otents can b e stated. The prime sp ectrum of b oth rings can also b e describ ed, and Dress’s c hara cterization o f solv able groups in terms of the connectedness of the sp e ctrum of the Burnside ring can b e generalized as well. F inally , b oth constructions hav e a natura l biset functor structure, fo r which they b e come Gr e en biset functors . A ma jor exception in this list of generalizable prop erties concerns unit 1 groups : it can b e show n that, unlike the case of the usual Burnside ring, the corresp ondence sending a finite g roup to the unit g r o up of either the slice or the section Burnside ring cannot b e endo we d with a structure of biset functor. This is due to the lack of a suitable tensor in d uction for these rings. The pap er is divided in tw o parts, and a n a pp endix : the first part is dev oted to the slic e Burnside functor. It consists of Sections 2 to 7. Section 2 recalls the basic definitions and pro p erties on the category of morphisms of G - sets. Section 3 in tro duce s the slice Burnside functor and it s Green biset functor structure. Section 4 is dev oted to the definition and main result on the g host map. In Section 5, the explicit formulas for t he primitive idemp oten ts of the slice Burnside algebra ov er Q are stated. Section 6 giv es a c hara cterization o f the image of the ghost map. Next Section 7 considers the prime sp ectrum of the slice Burnside ring. Finally , in Section 8, it is sho wn that tom Diec k’s t heorem, building o n D ress’s c haracterization of solv able groups, can b e extended to the slice Burnside ring : namely , F eit-Thompson’s theorem is equiv alen t to the fact that the only units in the slice Burnside ring of a group of o dd order are ± 1. But unlik e the case of the usual Burnside ring, the unit group of the slice Burnside ring cannot b e endo w ed with a structure of biset functor. The second part of the pap er is dev oted to the section Burnside ring, and it is organized similarly : Sec tion 9 in tro duces Galois morphisms, and states their general prop erties. In part icular, a left adjoin t functor to the forgetful functor from G alois morphisms to arbitra ry morphisms of G -sets is describ ed, whic h at t ac hes to an y morphism of G -sets a canonical G alois morphism. Section 10 considers the section Burnside functor, with its Green biset functor structure. Section 11 deals with the ghost map for the section Burnside ring. Sections 1 2 states the formulas fo r t he primitiv e idemp oten ts of the section Burnside Q -algebra, and Section 13 giv es a characterization of the ghost map. Section 14 considers the prime sp ectrum of t he section Burnside ring. In Section 15, the results ab out unit groups of Section 8 are extended to the unit group of the section Burnside ring . In particular, it is sho wn t hat this unit group cannot b e endo w ed with a structure of biset functor. The app endix deals with the functorial structure of the unit group of the slice Burnside ring and the section Burnside ring : it is p ossible to define biset functor op erations fo r these unit groups, but only for left i n ert bisets. This giv es tw o in teresting examples of someho w nat ura l biset functors without induction . The last result of the app endix is the explicit computation of this unit group in the case of an ab elian group. 2 I - The slic e Burnsid e ring 2. Morphisms of G -se t s 2.1. Definition : L et G b e a gr oup. If f : X → Y and f ′ : X ′ → Y ′ ar e mo rp hisms of G -se ts, a morphism fr om f to f ′ is a p air of morphisms of G -sets α : X → X ′ and β : Y → Y ′ such that the diagr am X f / / α   Y β   X ′ f ′ / / Y ′ is c ommutative. Morphisms of morphism s of G -se ts c an b e c omp os e d in the obvious w a y. This c omp o s ition endows the class of morphisms of G - s e ts with a structur e of c ate gory, den o te d by G - Mor . 2.2. Prop osi tion : The disjoint union of G -sets ind uc es a c opr o duct ( X f → Y , X ′ f ′ → Y ′ ) 7→ ( X ⊔ X ′ f ⊔ f ′ − → Y ⊔ Y ′ ) in the c ate gory G - Mor . Similarly, the dir e ct pr o duct o f G -sets, w ith diago n al G -a c tion, induc es a pr o duct ( X f → Y , X ′ f ′ → Y ′ ) 7→ ( X × X ′ f × f ′ − → Y × Y ′ ) in the c ate gory G - Mor . Pro of : F or any morphism of G -sets, the bijections Hom G - Set ( X ⊔ X ′ , A ) ∼ = Hom G - Set ( X , A ) × Hom G - Set ( X ′ , A ) induce ob vious bijections b et w een Hom G - Mor  ( X ⊔ X ′ ) f ⊔ f ′ − → ( Y ⊔ Y ′ ) , A α → B  and Hom G - Mo r ( X f → Y , A α → B ) × Ho m G - Mo r ( X ′ f ′ → Y ′ , A α → B ) . 3 These bijections are obv iously functorial in G - Mo r . Similarly , the bijections Hom G - Set ( A, X × X ′ ) ∼ = Hom G - Set ( A, X ) × Hom G - Set ( A, X ′ ) induce ob vious bijections b et w een Hom G - Mor  A α → B , ( X × X ′ ) f × f ′ − → ( Y × Y ′ )  and Hom G - Mor ( A α → B , X f → Y ) × Hom G - Mo r ( A α → B , X ′ f ′ → Y ′ ) . These bijections are obv iously functorial in G - Mo r . 3. The slice Burnsid e funct o r 3.1. Definition and Notation : L et G b e a gr oup. A slice of G is a p air ( T , S ) of sub gr oups of G w ith T ≥ S . A section of G is a slic e ( T , S ) with S E T . L et Π( G ) deno te the set of slic es of G , an d Σ( G ) denote the set of se ctions of G . When ( T , S ) ∈ Π( G ) , den ote by G/ S → G/T the pr o je ction morphism. 3.2. Definition : L et G b e a finite g r oup. The slice Burnside g roup Ξ( G ) of G is the quotient of the fr e e ab elian gr oup on the set of isomo rp h ism classes [ X f − → Y ] of morphism s of finite G -sets, by the sub gr oup gener ate d by elements of the form [( X 1 ⊔ X 2 ) f 1 ⊔ f 2 − → Y ] − [ X 1 f 1 → f ( X 1 )] − [ X 2 f 2 → f ( X 2 )] , whenever X f → Y is a mo rphism of finite G - s ets with a de c omp osition X = X 1 ⊔ X 2 as a disjoint union of G -sets, w her e f 1 = f | X 1 and f 2 = f | X 2 . When f : X → Y is a morphi s m of finite G -sets, let π ( f ) denote the image in Ξ( G ) of the isomorphism class of f . If S ≤ T ar e sub gr oups of G , set h T , S i G = π ( G/ S → G/T ) . 4 3.3. Lemma : 1. π ( ∅ → ∅ ) = 0 . 2. L et X f → Y b e a morphism of fin i te G -sets. Then π ( X f → Y ) = π  X f → f ( X )  . 3. L et X f → Y and X ′ f ′ → Y ′ b e mo rphisms of finite G -sets. Then π  ( X ⊔ X ′ ) f ⊔ f ′ − → ( Y ⊔ Y ′ )  = π ( X f → Y ) + π ( X ′ f ′ → Y ′ ) . Pro of : F or Assertion 1, set e = π ( ∅ → ∅ ). Since the morphism ∅ ⊔ ∅ → ∅ is isomorphic to ∅ → ∅ , it follow s that e + e = e , hence e = 0. F or Assertion 2, writing X = X ⊔ ∅ giv es π  X f → Y  = π  X f → f ( X )  + π ( ∅ → ∅ ) = π  X f → f ( X )  . F or Assertion 3, π  ( X ⊔ X ′ ) f ⊔ f ′ − → ( Y ⊔ Y ′ )  = π  X f → ( Y ⊔ Y ′ )  + π  X ′ f ′ → ( Y ⊔ Y ′ )  = π  X f → f ( X )  + π  X ′ f ′ → f ′ ( X ′ )  = π  X f → Y  + π  X ′ f ′ → Y ′  , where the first equalit y follows from the defining relations o f Ξ( G ), and the other ones from Assertion 2. 3.4. Lemm a : L e t f : X → Y b e a m orphism of finite G -sets. Then in the gr oup Ξ ( G ) π ( X f → Y ) = X x ∈ [ G \ X ] h G f ( x ) , G x i G Pro of : Indeed X ∼ = ⊔ x ∈ [ G \ X ] G/G x , and the image f ( G · x ) of the G -orbit of x is equal to the G - orbit of f ( x ). Moreov er the morphisms f | G · x : G · x → G · f ( x ) and G/G x → G/G f ( x ) are isomorphic. The claimed formula follows. 5 3.5. Cor o llary : The gr oup Ξ( G ) is gen e r ate d by the elements h T , S i G , wher e ( T , S ) runs thr o ugh a set [Π( G )] of r epr esentatives of c onjuga c y classes of slic es o f G . Pro of : Indeed, the morphisms G/ g S → G/ g T and G/ S → G/T a re isomor- phic, for any g ∈ G , and any slice ( T , S ) of G . 3.6. R emark : It will b e sho wn in Theorem 4.6 that this generating set is actually a basis of Ξ( G ). 3.7. Prop osition : The pr o d uct of morphisms ind uc es a c ommutative unital ring structur e on Ξ( G ) . Th e identity element for multiplic ation is the image of the class [ • → • ] , w her e • den otes a G -set of c ar dinality 1. Pro of : If w e can sho w that the pro duct of morphisms induces a w ell defined bilinear pro duct Ξ( G ) × Ξ( G ) → Ξ( G ), it will b e clear that this pro duc t is asso ciativ e, comm utativ e, and a dmits [ • → • ] as an iden tit y elemen t. Hence the only p oin t to chec k is that the pro duc t preserv es the defining relations of Ξ( G ). This is clear, since if g : Z → T is a morphism of finite G -sets, a nd if X 1 ⊔ X 2 f 1 ⊔ f 2 − → Y is a morphism, setting X = X 1 ⊔ X 2 , the domain of the morphism h : Z × X g × ( f 1 ⊔ f 2 ) / / T × Y has a disjoin t union decomp osition Z × X = ( Z × X 1 ) ⊔ ( Z × X 2 ), and moreo v er the restriction of g × f to Z × X 1 is g × f 1 . Th us π ( h ) = π  ( Z × X 1 ) g × f 1 − →  g ( Z ) × f 1 ( X 1 )   + π  ( Z × X 2 ) g × f 2 − →  g ( Z ) × f 2 ( X 2 )   = π  ( Z × X 1 ) g × f 1 − → ( T × f 1 ( X 1 ))  + π  ( Z × X 2 ) g × f 2 − → ( T × f 2 ( X 2 ))  where the last equalit y follows fro m Lemma 3.3 . 3.8. Prop osi tion : L et ( T , S ) and ( Y , X ) b e sli c es of G . Then in Ξ( G ) h T , S i G h Y , X i G = X g ∈ [ S \ G/X ] h T ∩ g Y , S ∩ g X i G . Pro of : Indeed ( G/S ) × ( G/X ) ∼ = ⊔ g ∈ [ S \ G/X ] G/ ( S ∩ g X ) , 6 via the map (from righ t to left) sending u ( S ∩ g X ) to ( uS, ug X ), for u ∈ G . The imag e of ( S, g X ) b y the map ( G/S ) × ( G/X ) → ( G/T ) × ( G/ Y ) is the pair ( T , g Y ), whose stabilizer in G is T ∩ g Y . The result no w follow s from Lemma 3.4. 3.9. Theorem : 1. L et G and H b e finite gr oups, and let U b e a finite ( H , G ) -biset. The functor ( X f → Y ) 7→ ( U × G X U × G f − → U × G Y ) fr om G - Mo r to H - Mo r induc es a gr o up homomorphi s m Ξ( U ) : Ξ( G ) → Ξ( H ) . 2. The c orr esp ondenc e G 7→ Ξ( G ) is a Gr e en biset functor. Pro of : F or Assertion 1, the only thing to chec k is that the defining relatio ns of Ξ( G ) are mapp ed to relations in Ξ( H ). But if X 1 ⊔ X 2 f 1 ⊔ f 2 − → Y is a morphism of finite G -sets, then U × G ( X 1 ⊔ X 2 ) ∼ = ( U × G X 1 ) ⊔ ( U × G X 2 ) . Moreo v er the image of t he map U × G f 1 is equal to U × G f 1 ( X 1 ). It follows that the relation [ X 1 ⊔ X 2 f 1 ⊔ f 2 − → Y ] − [ X 1 f 1 → f 1 ( X 1 )] − [ X 2 f 2 → f 2 ( X 2 )] in Ξ( G ) is mapp ed to the relation [ U X 1 ⊔ U X 2 U f 1 ⊔ U f 2 − → U Y ] − [ U X 1 U f 1 → ( U f 1 )( U X 1 )] − [ U X 2 U f 2 → ( U f 2 )( U X 2 )] , where U × G is abbreviated to U . It is no w clear that the corresp ondence sending a finite group G to Ξ( G ) and a finite ( H , G )-biset U to Ξ( U ) endo ws Ξ with a structure of biset functor (see [5]). Moreo v er if G , G ′ are finite groups, if f : X → Y is a morphism of finite G -sets and f ′ : X ′ → Y ′ is a mo r phism of finite G ′ -sets, then f × f ′ : X × X ′ → Y × Y ′ 7 is a morphism of G × G ′ -sets. This induces a pro duct Ξ( G ) × Ξ( G ′ ) → Ξ( G × G ′ ) , whic h is asso ciative in the ob vious sense. Moreo ver, the morphisms • → • of 1 -sets is obviously an identit y elemen t for this pro duct, up to iden tification G × 1 = G . Finally , if G , G ′ , H , H ′ are finite groups, if U is a finite ( H , G )-biset, if U ′ is a finite ( H ′ , G ′ )-biset, it is clear that the morphisms ( U × U ′ ) × G × G ′ ( f × f ′ ) and ( U × G f ) × ( U ′ × G ′ f ′ ) are isomorphic morphisms o f ( H × H ′ )-sets. Th us Ξ is a Green biset f unctor (see [5] Section 8.5). 3.10. Proposition : L et G and H b e finite gr oups, and U b e a finite ( H , G ) -bi s et. If ( T , S ) ∈ Π( G ) , then U × G h T , S i G = X u ∈ [ H \ U /S ] h u T , u S i H . (wher e u X = { h ∈ H | ∃ x ∈ X, hu = ux } , for X ≤ G ). Pro of : Indeed U × G ( G/S ) ∼ = U /S , a nd the stabilizer in H of uS is equal to u S . The result follo ws f r o m Lemma 3.4. The f ollo wing pro p osition clarifies the links existing b etw een the usual Burnside functor and the slice Burnside f unctor : 3.11. Prop osition : 1. L et G b e a finite gr oup. The c orr esp ondenc e se nding the morph ism X f → Y of finite G -sets to the G -set X ind uc es a unital ri n g homomor- phism s G fr om Ξ( G ) to the Burnside ring B ( G ) . 2. The c orr e sp ondenc e send ing the fi n ite G -se t X to the identity morphis m of X induc es a unital ring homomorphism i G : B ( G ) → Ξ( G ) , such that s G ◦ i G = Id B ( G ) . 3. As G varies, the morphism s s G and i G define morphism s of Gr e en b iset functors s : Ξ → B and i : B → Ξ , such that s ◦ i = Id B . In p articular i i s inje ctive, a n d s is surje ctive. Pro of : This is straightforw ard, fro m the definitions. 8 4. Slices and ghost map 4.1. Notation : L et S ≤ T b e sub gr o ups of G . I f X f → Y is a m orphism of finite G -sets, set φ T ,S ( X f → Y ) = | Hom G - Mor  G/S → G/T , X f → Y ) | . 4.2. Notation : Define a r elation  on the set Π( G ) by ( T , S )  ( Y , X ) ⇔ ( T ≤ Y and S ≤ X ) . The r elat io n  is an order relation on Π( G ). 4.3. Lemma : With this notation φ T ,S ( X f → Y ) = | f − 1 ( Y T ) S | . In p articular, fo r any A ≤ B ≤ G , φ T ,S ( G/ A → G/B ) = |{ g ∈ G/ A | ( T g , S g )  ( B , A ) }| . Pro of : The mo r phisms of G - sets f r o m G/ S to X a re in one to one corresp on- dence with the set X S of fixed p oin ts o f S on X : the morphism asso ciated to x ∈ X S is defined by g S 7→ g x . Similarly , the ho mo mo r phisms of G -sets from G/T to Y a re in o ne to one corresp ondence with Y T . Hence the set Hom G - Mo r ( G/S → G/T , X f → Y ) is in one t o one corresp ondence with the set of pairs ( x, y ) ∈ X S × Y T suc h that f ( x ) = y , i.e. with the elemen ts x of f − 1 ( Y T ) S . 4.4. Corollary : L et ( T , S ) ∈ Π( G ) and p b e a prim e numb er. If P is a p -sub gr oup of N G ( T , S ) , then φ T ,S ≡ φ P T ,P S (mo d . p ) . 9 Pro of : Indeed fo r an y morphism of finite G -sets X f → Y , the set f − 1 ( Y T ) S is in v aria n t b y N G ( T , S ), thus, as P is a p -g roup, | f − 1 ( Y T ) S | ≡ | f − 1 ( Y T ) P S | (mo d . p ) , and moreo v er f − 1 ( Y T ) P S = f − 1 ( Y P T ) P S . 4.5. Prop osition : L et S ≤ T b e sub gr oups of G . Then the map φ T ,S induc es a ring homomorp hism Ξ( G ) → Z , stil l deno te d by φ T ,S . Pro of : Since the pro duct of morphisms ( X f → Y , X ′ f ′ → Y ′ ) 7→  ( X × X ′ ) f × f ′ − → ( Y × Y ′ )  is a pro duct in the category G - M o r , it follows that φ T ,S  ( X × X ′ ) f × f ′ − → ( Y × Y ′ )  = φ T ,S ( X f → Y ) φ T ,S ( X ′ f ′ → Y ′ ) . Also φ T ,S ( • → • ) = 1. The only thing to c hec k is that φ T ,S induces a we ll defined map Ξ( G ) → Z , i.e. that the defining relations of Ξ( G ) are mapp ed to 0 b y φ T ,S . First, by Lemma 4.3, for any morphism o f G - sets f : X → Y φ T ,S ( X f → Y ) = φ T ,S  X f → f ( X )  . No w φ T ,S  ( X 1 ⊔ X 2 ) f 1 ⊔ f 2 − → Y  = | ( f 1 ⊔ f 2 ) − 1 ( Y T ) S | = | X 1 ∩ ( f 1 ⊔ f 2 ) − 1 ( Y T ) S | + | X 2 ∩ ( f 1 ⊔ f 2 ) − 1 ( Y T ) S | = | f − 1 1 ( Y T ) S | + | f − 1 2 ( Y T ) S | = φ T ,S ( X 1 f 1 → Y ) + φ T ,S ( X 2 f 2 → Y ) = φ T ,S  X 1 f 1 → f ( X 1 )  + φ T ,S  X 2 f 2 → f ( X 2 )  . This completes the pro of. 4.6. Theorem : The gr oup Ξ( G ) is a fr e e ab elian gr oup, with b asis the set of elements h T , S i G , wher e ( T , S ) runs thr ough a se t [Π( G )] of r epr esentatives of c onjugacy classes of slic es o f G . Mor e ove r, the map (c al le d the ghost map for Ξ( G ) ) Φ = Y ( T ,S ) ∈ [Π( G )] φ T ,S : Ξ( G ) → Y ( T ,S ) ∈ [Π( G )] Z is an i n je ctive ring homomorphism , with finite c okern e l a s morphism of ab elian gr oups. 10 Pro of : By Lemma 3.5, the elemen ts h T , S i G , for ( T , S ) ∈ [Π( G )], generate Ξ( G ). Supp ose that there is a non zero linear com binatio n in the k ernel of Φ Λ = X ( T ,S ) ∈ [Π( G )] λ T ,S h T , S i G with in teger co effi cien ts λ T ,S ∈ Z , f or ( T , S ) ∈ [Π( G )]. Extend λ to a function Π( G ) → Z , constan t on conjugacy classes. Let ( Y , X ) b e an elemen t of Π( G ), maximal for the relation  , such that λ Y , X 6 = 0. Then since b y Lemma 4.3 φ Y , X ( G/S → G/T ) = |{ g ∈ G/S | ( Y g , X g )  ( T , S ) }| , it follow s that φ Y , X (Λ) = X ( T ,S ) ∈ [Π( G )] λ T ,S φ Y , X ( G/S → G/T ) = λ Y , X φ Y , X ( G/X → G/ Y ) = λ Y , X | N G ( X , Y ) / X | = 0 . Hence λ Y , X = 0, and this con tradiction shows that Φ is inj ective . In particu- lar, the elemen ts h T , S i G , for ( T , S ) ∈ [Π( G )], form a Z -basis of Ξ( G ). Thus Φ is an injectiv e morphism b e t w een free a b elian gro ups with the same finite rank, hence it has finite cok ernel. 4.7. Corollary : Set Q Ξ( G ) = Q ⊗ Z Ξ( G ) , an d Q Φ = Q ⊗ Z Φ . Then Q Φ : Q Ξ( G ) → Y ( T ,S ) ∈ [Π( G )] Q is an isom orphism of Q -algebr as. 5. Slices and idemp oten ts By Corollary 4.7, the comm utative Q -algebra Q Ξ( G ) is split semisimple. Its primitiv e idemp otents are indexed by slices of G , up to conjugatio n : they are the in v erse images under Q Φ of the primitiv e idemp ot ents of the algebra Q ( T ,S ) ∈ [Π( G )] Q : 11 5.1. Notation : If ( T , S ) ∈ Π( G ) , d enote by ξ G T ,S the unique element of Q ⊗ Z Ξ( G ) such that ∀ ( Y , X ) ∈ Π( G ) , Q φ Y , X ( ξ G T ,S ) =  1 if ( Y , X ) = G ( T , S ) 0 otherwi s e The set of el e ments ξ G T ,S , for ( T , S ) ∈ [Π( G )] , is the set of primitive idemp o- tents of Q Ξ( G ) . 5.2. Theorem : L et ( T , S ) ∈ Π( G ) . Then ξ G T ,S = 1 | N G ( T , S ) | X ( V ,U )  ( T ,S ) | U | µ Π  ( V , U ) , ( T , S )  h V , U i G , wher e µ Π is the M¨ obius function of the p oset (Π( G ) ,  ) . Pro of : Denote b y Π( G ) the set of orbits of G f or it s conjuga cy a ctio n on Π( G ). Th us Π( G ) is in one t o one cor r esp o ndence with [Π( G )], and the map Q Φ can a lso b e view ed a Q -alg ebra isomorphism f rom Q Ξ( G ) to Q Π( G ) . The Q -v ector space Q Ξ( G ) has a basis consisting of the elemen ts h V , U i G , for ( V , U ) ∈ [Π( G )]. Let Q Π( G ) denote the Q -v ector space with ba sis Π( G ), and let p : Q Π( G ) → Q Ξ( G ) denote the Q -linear map sending ( V , U ) ∈ Π( G ) to h V , U i G . Let β G T ,S denote the v ector of the canonical basis of Q Π( G ) indexed by the G -orbit o f ( T , S ) ∈ Π( G ), and let q : Q Π( G ) → Q Π( G ) denote the Q - linear map sending ( T , S ) ∈ Π( G ) to β G T ,S . With this notation, Lemma 4.3 show s that for any ( V , U ) ∈ Π( G ) Φ ◦ p  ( V , U )  = X ( T ,S ) ∈ [Π( G )] φ T ,S ( h V , U i G ) β G T ,S = X ( T ,S ) ∈ [Π( G )] 1 | U |   { g ∈ G | ( T , S )  ( g V , g U ) }   β G T ,S = X ( T ,S ) ∈ Π( G ) | N G ( T , S ) | | G || U |   { g ∈ G | ( T g , S g )  ( V , U ) }   β G T ,S = X ( T ,S ) ∈ Π( G ) g ∈ G ( T g ,S g )  ( V ,U ) | N G ( T g , S g ) | | G || U | β G T g ,S g 12 Th us Φ ◦ p  ( V , U )  = X ( T ,S ) ∈ Π( G ) ( T ,S )  ( V ,U ) | N G ( T , S ) | | U | β G T ,S This sho ws that if e Φ is the Q -endomorphism of Q Π( G ) defined b y e Φ  ( V , U )  = X ( T ,S ) ∈ Π( G ) ( T ,S )  ( V ,U ) | N G ( T , S ) | | U | ( T , S ) , then Φ ◦ p = q ◦ e Φ. The matrix of the map e Φ is equal to the pro duct E · J · D , where D is a diagona l matrix with dia gonal co effi cien ts ( | N G ( T , S ) | ) ( T ,S ) ∈ Π( G ) , where E is a diagonal ma t r ix with diagonal co effi cien t s ( 1 | U | ) ( V ,U ) ∈ Π( G ) , and J is the incidence matrix of the order relation  on Π( G ). It follow s that e Φ is in v ertible, with in vers e equal to D − 1 · J − 1 · E − 1 . No w the en t r ies of the matrix J − 1 are precisely the v alues of the M¨ obius function µ Π of the p oset  Π( G ) ,   . It follo ws that ξ G T ,S = Φ − 1 ( β G T ,S ) = Φ − 1 ◦ q  ( T , S )  = p ◦ e Φ − 1  ( T , S )  = 1 | N G ( T , S ) | X ( V ,U ) ∈ Π( G ) ( V ,U )  ( T ,S ) | U | µ Π  ( V , U ) , ( T , S )  h V , U i G , whic h completes the pro of . 5.3. Prop osi tion : L et ( X , ≤ ) b e a fini te p ose t. L et Π( X ) denote the set of p airs ( y , x ) of elements of X such that x ≤ y . D efine a p artial or der  on Π( X ) by ∀ ( y , x ) , ( t, z ) ∈ Π( X ) , ( y , x )  ( t, z ) ⇔ y ≤ t and x ≤ z . Then the M¨ obius function µ Π of the p oset (Π( X ) ,  ) c an b e c omp ute d as fol lows, for any ( y , x ) , ( t, z ) ∈ Π( X ) : ( 5 . 4 ) µ Π  ( y , x ) , ( t, z )  =  µ X ( x, z ) µ X ( y , t ) if x ≤ z ≤ y ≤ t 0 otherwise , wher e µ X is the M¨ obius function of the p oset ( X , ≤ ) . 13 Pro of : Let m  ( y , x ) , ( t, z )  denote the expression defined b y the right hand side of Equation 5.4. Then if ( y , x )  ( t, z ), i.e. if y ≤ t and x ≤ z , X ( v,u ) ∈ Π( X ) ( y, x )  ( v,u )  ( t,z ) m  ( y , x ) , ( v , u )  = X y ≤ v ≤ t x ≤ u ≤ z u ≤ v, u ≤ y µ X ( x, u ) µ X ( y , v ) = X y ≤ v ≤ t x ≤ u ≤ z u ≤ y µ X ( x, u ) µ X ( y , v ) =  X y ≤ v ≤ t µ X ( y , v )  X x ≤ u ≤ z u ≤ y µ X ( x, u )  . The first f actor P y ≤ v ≤ t µ X ( y , v ) is equal to 0 if y 6 = t , and to 1 if y = t . In this case, the second factor P x ≤ u ≤ z u ≤ y µ X ( x, u ) is equal to P x ≤ u ≤ z µ X ( x, u ). This is equal to zero if x 6 = z , a nd to 1 if x = z . It follo ws that P ( v,u ) ∈ Π( X ) ( y, x )  ( v,u )  ( t,z ) m  ( y , x ) , ( v , u )  is equal to 0 if ( y , x ) 6 = ( t, z ) and to 1 otherwise. The prop o sition follow s. Applying Prop osition 5.3 to the po set of subgroups of G , ordered b y inclusion of subgroups, give s the following : 5.5. Corollary : L et ( V , U ) and ( T , S ) b e slic es of G . Then ( 5 . 6 ) µ Π  ( V , U ) , ( T , S )  =  µ ( U, S ) µ ( V , T ) if U ≤ S ≤ V ≤ T 0 otherwise , wher e µ is the M¨ obius function of the p oset of sub gr oups of G . In p articular in Ξ( G ) ξ G T ,S = 1 | N G ( T , S ) | X U ≤ S ≤ V ≤ T | U | µ ( U, S ) µ ( V , T ) h V , U i G . 6. The image of the g h ost map The following c hara cterization of the imag e of the ghost map is the a nalogue for the slice Burnside ring of a theorem of D ress ([6]) on the ordinary Burnside 14 ring : 6.1. Theorem : L et G b e a finite gr o up, and let m = ( m T ,S ) ( T ,S ) ∈ Π( G ) b e a se q uenc e of inte g e rs indexe d by Π( G ) , c onstant o n G -c onjugacy cl a sses of slic es. Then the se quenc e [ m ] = ( m T ,S ) ( T ,S ) ∈ [Π( G )] of r epr ese n tatives lies in the image of the ghost map Φ if and only if, fo r any slic e ( T , S ) of G X g ∈ N G ( T ,S ) /S m , ≡ 0  mo d . | N G ( T , S ) /S |  . Pro of : Sa ying that [ m ] lies in the image of Φ is equiv alen t to say ing that the elemen t s m = X ( T ,S ) ∈ [Π( G )] m T ,S ξ G T ,S of Q Ξ( G ) lies in Ξ( G ), i.e. that it is a linear com bination with in teger co efficien ts of t he elemen ts h V , U i G , for ( V , U ) ∈ [Π( G )]. Now by Theorem 5.2 s m = X ( T ,S ) ∈ Π( G ) | N G ( T , S ) | | G | m T ,S ξ G T ,S = 1 | G | X ( T ,S ) ∈ Π( G ) m T ,S X ( V ,U )  ( T ,S ) | U | µ Π  ( V , U ) , ( T , S )  h V , U i G = 1 | G | X ( V ,U ) ∈ Π( G ) | U |  X ( V ,U )  ( T ,S ) µ Π  ( V , U ) , ( T , S )  m T ,S  h V , U i G = X ( V ,U ) ∈ [Π( G )] 1 | N G ( V , U ) /U |  X ( V ,U )  ( T ,S ) µ Π  ( V , U ) , ( T , S )  m T ,S  h V , U i G . Hence s m ∈ Im Φ if and only if the num b er β V , U = 1 | N G ( V , U ) /U |  X ( V ,U )  ( T ,S ) µ Π  ( V , U ) , ( T , S )  m T ,S  is an in teger, for any slice ( V , U ) of G . With this notation, for an y ( Y , X ) ∈ Π( G ) m Y , X = X ( V ,U ) ∈ Π( G ) ( Y ,X )  ( V ,U ) β V , U | N G ( V , U ) /U | . 15 Hence, setting σ Y , X = P g ∈ N G ( Y ,X ) /X m , : σ Y , X = X g ∈ N G ( Y ,X ) /X X ( V ,U ) ∈ Π( G ) ( , )  ( V ,U ) β V , U | N G ( V , U ) /U | = X g ∈ N G ( Y ,X ) ( V ,U ) ∈ Π( G ) ( , )  ( V ,U ) 1 | X | β V , U | N G ( V , U ) /U | = X ( V ,U ) ∈ Π( G ) ( Y ,X )  ( V ,U ) g ∈ U ∩ N G ( Y ,X ) 1 | X | β V , U | N G ( V , U ) /U | σ Y , X = X ( V ,U ) ∈ [ N G ( Y ,X ) \ Π( G )] ( Y ,X )  ( V ,U ) | N G ( Y , X ) /X | | N G ( Y , X , V , U ) | β V , U | N G ( V , U ) /U || U ∩ N G ( Y , X ) | = | N G ( Y , X ) /X | X ( V ,U ) ∈ [ N G ( Y ,X ) \ Π( G )] ( Y ,X )  ( V ,U ) β V , U | N G ( V , U ) || U ∩ N G ( Y , X ) | | U || N G ( Y , X , V , U ) | = | N G ( Y , X ) /X | X ( V ,U ) ∈ [ N G ( Y ,X ) \ Π( G )] ( Y ,X )  ( V ,U ) β V , U | N G ( V , U ) : U · N G ( Y , X , V , U ) | . Setting α Y , X = σ Y , X / | N G ( Y , X ) /X | , it follows that α Y , X = X ( V ,U ) ∈ N G ( Y ,X ) \ Π( G ) ( Y ,X )  ( V ,U ) β V , U | N G ( V , U ) : U · N G ( Y , X , V , U ) | . Since | N G ( V , U ) : U · N G ( Y , X , V , U ) | = 1 if ( Y , X ) = ( V , U ), it f o llo ws that the tra nsition mat r ix from t he β V , U ’s to the α Y , X ’s is triangular, with in teger co efficien ts, a nd 1’s on the diagonal. Hence it is in v ertible o v er Z , and the n umbers β V , U are all integers, fo r ( V , U ) ∈ Π( G ), if and only if the num b ers α Y , X are a ll in tegers, for ( Y , X ) ∈ Π( G ). This completes the pro of . 7. Prime sp ectrum 7.1. Notation : L et p den ote either 0 or a pri m e numb e r. 16 • If ( T , S ) ∈ Π( G ) , let I T ,S,p b e the prime i d e al of Ξ( G ) define d as the kernel of the ring homomorphis m Ξ( G ) φ T ,S − → Z → Z /p Z , wher e the right hand side map is the pr oje ction. • L et Θ( G ) denote the set of triples ( T , S, p ) , wher e ( T , S ) ∈ Π( G ) is such that | N G ( T , S ) /S | 6≡ 0 (mo d . p ) . The group G acts on Θ( G ), by g ( T , S, p ) = ( g T , g S , p ), for g ∈ G , and the ideal I T ,S,p only dep ends on the G -orbit of ( T , S, p ) . Con v ersely : 7.2. Proposit ion : L et I b e a prim e ide al of Ξ( G ) , and R = Ξ( G ) /I . Denote by φ : Ξ( G ) → R the pr o je ction m ap, and denote by p ≥ 0 the char acteristic of R . Then R ∼ = Z /p Z a nd : 1. If p = 0 , ther e exists a slic e ( T , S ) of G such that φ = φ T ,S , and ( T , S ) is unique up to G -c onjugation, with this pr op erty. 2. If p > 0 , ther e exists a slic e ( T , S ) of G s uch that φ is the r e duction mo dulo p o f φ T ,S and N G ( T , S ) /S is a p ′ -gr oup, and ( T , S ) is unique up to G -c o n jugation, wi th these pr op erties. In p articular, ther e exists a unique ( T , S, p ) ∈ Θ( G ) , up to c onjugation, such that I = I T ,S,p . Pro of : Let ( T , S ) b e a slice of G , minimal for the relation  , suc h that h T , S i G / ∈ I . Then by Prop o sition 3.8, for an y ( Y , X ) ∈ Π( G ) h T , S i G h Y , X i G = X g ∈ [ S \ G/X ] h T ∩ g Y , S ∩ g X i G ≡ X g ∈ G/X S ≤ g X T ≤ g Y h T , S i G (mo d . I ) = φ T ,S  h Y , X i G  h T , S i G . Since I is prime, it follo ws that h Y , X i G − φ T ,S ( h Y , X i G )1 Ξ( G ) ∈ I . In partic- ular R = Ξ( G ) / I is generated b y t he image of 1 Ξ( G ) , hence R ∼ = Z /p Z , where p is the characteristic of R . Since R is an in tegral domain, the n umber p is either 0 or a prime. 1. If p = 0, then R = Z , and φ = φ T ,S . And if ( T ′ , S ′ ) ∈ Π( G ) is suc h that φ T ,S = φ T ′ ,S ′ , then b oth φ T ,S  h T ′ , S ′ i G  and φ T ′ ,S ′  h T , S i G  are non 17 zero. Then there exist elemen ts g , g ′ ∈ G suc h that ( T g , S g )  ( T ′ , S ′ ) and ( T ′ g ′ , S ′ g ′ )  ( T , S ), so ( T , S ) and ( T ′ , S ′ ) a re conjugate in G . 2. If p > 0, then R = Z /p Z , and φ is equal to the reduction of φ T ,S mo dulo p . Since φ  h T , S i G  = | N G ( T , S ) /S | is non zero in R , it follows that N G ( T , S ) /S is a p ′ -group. If ( T ′ , S ′ ) is another slice of G suc h that φ is the reduction mo dulo p o f φ T ′ ,S ′ , and N G ( T ′ , S ′ ) /S ′ is a p ′ -group, then | N G ( T , S ) /S | = φ T ,S  h T , S i G  ≡ φ T ′ ,S ′  h T , S i G  (mo d . p ) . This is non zero. Similarly | N G ( T , S ) /S | ≡ φ T ,S  h T ′ , S ′ i G  (mo d . p ) is non zero. In particular φ T ′ ,S ′  h T , S i G  and φ T ,S  h T ′ , S ′ i G  are b oth non zero, and it follows as ab o v e that ( T , S ) and ( T ′ , S ′ ) are conjugate in G . 7.3. Notation : L et p b e a prime numb er. • L et Π p ( G ) denote the subset of Π( G ) c onsi s ting of the slic e s ( T , S ) such that N G ( T , S ) /S is a p ′ -gr oup. • F or any ( T , S ) ∈ Π( G ) , le t ( T , S ) b p denote the unique elemen t ( V , U ) of Π p ( G ) , up to c onjugation, such that I T ,S,p = I V , U,p . 7.4. Prop osition : L et p b e a prime numb e r. If ( T , S ) is a slic e of G , let ( T , S ) + p denote a slic e of the fo rm ( P T , P S ) of G , w h er e P is a Sylow p -sub gr oup of N G ( T , S ) . Define inductively an inc r e asing se quenc e ( T n , S n ) in (Π( G ) ,  ) by ( T 0 , S 0 ) = ( T , S ) , and ( T n +1 , S n +1 ) = ( T n , S n ) + p , fo r n ∈ N . Then ( T , S ) b p is c onjugate to the lar gest term ( T ∞ , S ∞ ) of the se quenc e ( T n , S n ) . Pro of : The pro of of Prop osition 7.2 sho ws that ( T , S ) b p is a minimal elemen t ( V , U ) of t he p oset ( Π( G ) ,  ) suc h that φ T ,S ( V , U ) = |{ g ∈ G/U | ( T g , S g )  ( V , U ) }| 6≡ 0 (mo d .p ) . Th us one can assume that ( T , S )  ( V , U ). But φ T ,S ≡ φ P T ,P S (mo d . p ) b y Corollary 4.4, f or any p -subgroup P of N G ( T , S ), hence one can also a ssume that ( T , S ) + p  ( V , U ), and b y induction, that ( T ∞ , S ∞ )  ( V , U ). Moreov er φ T ∞ ,S ∞ ≡ φ V , U (mo d . p ). As N G ( T ∞ , S ∞ ) /S ∞ is a p ′ -group, it follow s that ( T ∞ , S ∞ ) = ( V , U ), as w as to b e sho wn. 7.5. Remark : Let ( T , S ) ∈ Π( G ), and ( V , U ) ∈ Π p ( G ). It is easy to c hec k, b y induction o n the integer n suc h t ha t ( T n , S n ) = ( T ∞ , S ∞ ), t ha t ( T , S ) b p 18 is equal to ( V , U ) if and only if T is a subnormal subgroup of V , if S is a subnormal subgro up of U , if | U : S | is a p o w er of p , and if the set T · U is equal to V . 7.6. Prop osit ion : L et ( T , S, p ) , ( T ′ , S ′ , p ′ ) b e elements of Θ ( G ) . Then I T ′ ,S ′ ,p ′ ⊆ I T ,S,p if and onl y if • either p ′ = p and the slic es ( T ′ , S ′ ) and ( T , S ) ar e c onjugate in G . • or p ′ = 0 and p > 0 , a n d the slic es ( T ′ , S ′ ) b p and ( T , S ) ar e c onjugate in G . Pro of : Set I = I T ,S,p and I ′ = I T ′ ,S ′ ,p ′ . Then Ξ( G ) /I ′ ∼ = Z /p ′ Z maps surjectiv ely to Ξ( G ) / I ∼ = Z /p Z . Th us if p = p ′ , this pro jection map is an isomorphism, hence I = I ′ and the slices ( T , S ) and ( T ′ , S ′ ) are conjugate in G . And if p 6 = p ′ , then p ′ = 0 and p > 0. The morphism φ T ,S is equal to the reduction mo dulo p of the morphism φ T ′ ,S ′ . In other w ords I T ′ ,S ′ ,p = I T ,S,p , hence ( T , S ) is conjug a te to ( T ′ , S ′ ) b p . 7.7. Cor ollary : L et p b e a prime n umb er, and let Z ( p ) b e the lo c ali z a tion of Z at the set Z − p Z . L e t Θ p ( G ) denote the subset of Θ( G ) c onsisting o f triples ( T , S, 0) , fo r ( T , S ) ∈ Π( G ) , and ( T , S, p ) , fo r ( T , S ) ∈ Π p ( G ) . Then : 1. The prime ide als of the ring Z ( p ) Ξ( G ) ar e the ide als Z ( p ) I T ,S,q , for ( T , S, q ) ∈ Θ p ( G ) . 2. If ( T , S, q ) , ( T ′ , S ′ , q ′ ) ∈ Θ p ( G ) , then Z ( p ) I T ′ ,S ′ ,q ′ ⊆ Z ( p ) I T ,S,q if and only if : • either q = q ′ , and the slic es ( T , S ) and ( T ′ , S ′ ) ar e c onjugate in G . • or q ′ = 0 , q = p , and the slic es ( T ′ , S ′ ) b p and ( T , S ) ar e c onjugate in G . 3. The c onne cte d c omp onents of the sp e ctrum of Z ( p ) Ξ( G ) ar e indexe d by the c onjugacy classes of Π p ( G ) . The c o m p onent ind e xe d by ( T , S ) ∈ Π p ( G ) c onsists of a unique ma x imal element Z ( p ) I T ,S,p , and of the ide als Z ( p ) I T ′ ,S ′ , 0 , wher e ( T ′ , S ′ ) ∈ Π( G ) is such that ( T ′ , S ′ ) b p is c onjugate to ( T , S ) in G . Pro of : The pr ime ideals of Z ( p ) Ξ( G ) are of the form Z ( p ) I , where I is a prime ideal of Ξ( G ) suc h that I ∩ ( Z − p Z ) = ∅ . Equiv alently I = I T ,S, 0 or I = I T ,S,p . This prov es Assertion 1. No w Assertion 2 follow s from Prop osition 7.6, and Assertion 3 follows from Assertion 2. 19 7.8. Corollary : 1. The primitive idemp otents o f the ring Z ( p ) Ξ( G ) ar e inde x e d by the c on- jugacy classes of Π p ( G ) . The primitive i demp otent η G V , U indexe d by ( V , U ) ∈ Π p ( G ) is e qual to η G V , U = X ( T ,S ) ∈ [Π( G )] ( T ,S ) b p = G ( T ,S ) ξ G T ,S . 2. L et π b e a set of prime numb ers, and Z ( π ) b e the lo c alization of Z r elative to Z − ∪ p ∈ π p Z . L et F b e a set of slic es of G , in v ariant by G - c onjugation, and [ F ] b e a set of r e pr esentatives of G - c onjugacy classes of F . Then the fol lowing c onditions ar e e q uivalent : (a) The idemp otent ξ G F = X ( T ,S ) ∈ [ F ] ξ G T ,S of Q Ξ( G ) lies in Z ( π ) Ξ( G ) . (b) L et ( T , S ) ∈ Π( G ) , and let P b e a p -sub gr oup of N G ( T , S ) , for some p ∈ π . Then ( T , S ) ∈ F if a nd only if ( P T , P S ) ∈ F . Pro of : Let F b e a set of slices o f G , inv ariant by G - conjugation, and [ F ] b e a set of represen tat ives of G -conjuga cy classes of F . The idempotent ξ G F = X ( T ,S ) ∈ [ F ] ξ G T ,S of Q Ξ( G ) lies in Z ( p ) Ξ( G ), for some prime p , if and only if there exists a n in t eger m , not divisible b y p , suc h that u = mξ G F ∈ Ξ( G ). L et ( T , S ) ∈ Π( G ), and let P b e a p -subgroup of N G ( T , S ). The inte ger φ T ,S ( u ) is equal to m if ( T , S ) ∈ F , and to 0 otherwise. Hence it is coprime to p if a nd only if ( T , S ) ∈ F . Since φ T ,S and φ P T ,P S are congruent mo dulo p , it fo llo ws that ( T , S ) ∈ F if and only if ( P T , P S ) ∈ F . Hence if ( T , S ) and ( T ′ , S ′ ) are slices of G suc h that ( T , S ) b p = G ( T ′ , S ′ ) b p , then ( T , S ) ∈ F if and only if ( T ′ , S ′ ) ∈ F . Thus F is a disjoint union of sets of the form E V , U = { ( T , S ) ∈ Π( G ) | ( T , S ) b p = G ( V , U ) } , for some slices ( V , U ) ∈ Π p ( G ). In other w ords the idemp o ten t ξ G F is a sum of some idemp oten ts η G V , U , for ( V , U ) ∈ Π p ( G ). 20 But the primitive idemp otents of the ring Z ( p ) Ξ( G ) are in one to one cor- resp ondence with the connected comp onen ts of its sp ectrum, whic h precisely are indexed by the conjuga cy classes of Π p ( G ). It follows that ξ G F = η G V , U is equal to the idemp oten t corresp onding to the comp onen t indexed b y ( V , U ), for any ( V , U ) ∈ Π p ( G ). This pro v es Assertion 1. This also pro v es Assertion 2 in the case where π consists of a single prime n umber. F or t he general case, observ e that ξ G F lies in Z ( π ) Ξ( G ) if and only if it lies in Z ( p ) Ξ( G ), for any p ∈ π . 7.9. Theorem : L et G b e a finite gr oup. 1. L et ∼ denote the finest e quivalenc e r elation on the set Π( G ) such that for any ( T , S ) , ( T ′ , S ′ ) ∈ Π( G ) , ∃ p, ( T , S ) b p = G ( T ′ , S ′ ) b p = ⇒ ( T , S ) ∼ ( T ′ , S ′ ) . Then the p ri m itive idemp otents of Ξ( G ) ar e indexe d by the e quivalenc e classes of Π( G ) for the r elation ∼ . The idemp otent ξ G C indexe d by the c omp onent C is e qual to ξ G C = X ( T ,S ) ∈ [ C ] ξ G T ,S , wher e [ C ] is a set of r epr e s e ntatives of G -c on jugacy classe s in C . 2. The prime sp e ctrum o f Ξ( G ) is c onne cte d if and on ly if G is solvable. Pro of : Assertion 1 f ollo ws fro m Corollary 7.8, applied to the set π of a ll primes. F or Assertion 2, observ e t ha t the sp ectrum of Ξ( G ) is connected if and only if 1 is a primitive idemp o ten t of Ξ( G ). This means that for an y t wo slices ( T , S ) and ( T ′ , S ′ ) of G , there exists a sequence ( T i , S i ) of slices, for i ∈ { 0 , . . . , n } , a nd a sequence p i of prime num b ers, for i ∈ { 0 , . . . , n − 1 } , suc h that ( T , S ) = ( T 0 , S 0 ) p 0 ∼ ( T 1 , S 1 ) p 1 ∼ . . . p n − 2 ∼ ( T n − 1 , S n − 1 ) p n − 1 ∼ ( T n , S n ) = ( T ′ , S ′ ) , where the notation ( Y , X ) p ∼ ( Y ′ , X ′ ) means that ( Y , X ) b p = G ( Y ′ , X ′ ) b p . But clearly , if ( Y , X ) p ∼ ( Y ′ , X ′ ), then t he slices  O p ( Y ) , O p ( X )  and  O p ( Y ′ ) , O p ( X ′ )  are conjug a te in G . Hence the slices  D ∞ ( Y ) , D ∞ ( X )  and  D ∞ ( Y ′ ) , D ∞ ( X ′ )  are conjugate in G , where D ∞ ( H ) denotes the last term in the deriv ed series of the group H . 21 If the sp ectrum of G is connected, ta king ( T , S ) = ( G, G ) and ( T ′ , S ′ ) = ( 1 , 1 ), it follows that D ∞ ( G ) = 1 , i.e. that G is solv able. Con vers ely , if G is solv able, let ( T , S ) b e a slice of G . Then S is solv able, and there exis ts a prime p suc h that O p ( S ) < S . Let P be a Sylow p - subgroup of S . Then P ≤ N G ( T , S ), and ( T , S ) =  P T , P O p ( S )  . Th us ( T , S ) p ∼  T , O p ( S )  . By induction, there is a sequence of prime n umbers p i , for i ∈ { 1 , . . . , k } , suc h that ( T , S ) = ( T , S 0 ) p 0 ∼ ( T , S 1 ) p 1 ∼ . . . p k − 1 ∼ ( T , S k ) p k ∼ ( T , 1 ) , where S i +1 = O p i ( S i ). No w T is solv able, so there exists a prime q suc h that O q ( T ) < T . If Q is a Sylow q -subgroup of T , then Q ≤ N G ( T , 1 ), and ( T , 1 ) q ∼  O q ( T ) , 1  . Hence there exists a sequence of primes q j , for j ∈ { 0 , . . . , l } , suc h that ( T , 1 ) = ( T 0 , 1 ) q 0 ∼ ( T 1 , 1 ) q 1 ∼ . . . q l − 1 ∼ ( T l , 1 ) q l ∼ ( 1 , 1 ) , where T j +1 = O q j ( T j ). This sho ws that the sp e ctrum of Ξ( G ) is connected. 8. Unit group 8.1. When G is a finite group, denote b y Ξ( G ) × the group of in v ertible elemen ts of the ring Ξ( G ). It follows from Theorem 4 .6 that the restricted ghost map yields an injectiv e gro up homomorphism Φ × : Ξ( G ) × ֒ → Y ( T ,S ) ∈ Π( G ) Z × . The follo wing lemma is a straigh tforw ard conse quence of the existence of this injectiv e group homomorphism : 8.2. Lemma : L et G b e a fi nite gr oup, and le t u ∈ Ξ( G ) . The fol lowing c onditions ar e e quival e nt : 1. u ∈ Ξ( G ) × . 2. φ T ,S ( u ) ∈ {± 1 } , for any ( T , S ) ∈ Π( G ) . 3. u 2 = 1 . In p articular Ξ( G ) × is a finite elemen tary ab elian 2-gr oup. The main motiv ation in considering the g r o up Ξ( G ) × lies in the following prop osition, whic h extends a theorem of tom Diec k ( [9 ] Prop osition 1.5.1) ab out the unit group o f the usual Burnside r ing : 22 8.3. Pr op osition : F eit-Th o m pson ’s the or em is e quivalent to the s tatemen t that, i f G h a s o dd or der, then Ξ( G ) × = {± 1 } . Pro of : The first o bserv ation is that fo r a n y finite group G , by Lemma 8.2 , the corresp ondences u 7→ 1 − u 2 and e 7→ 1 − 2 e are m utually inv erse bijections b et w een Ξ( G ) × and the set of idemp otents e ∈ Q Ξ( G ) such that 2 e ∈ Ξ( G ). No w Theorem 5.2 sho ws that | G | e ∈ Ξ( G ), for a ny idempot ent e of Q Ξ( G ). Hence if G has o dd order, and if e is an idemp otent of Ξ( G ) suc h that 2 e ∈ Ξ( G ), then (2 , | G | ) e = e ∈ Ξ( G ). Th us if | G | is o dd, the set Ξ( G ) × is in one to one corresp o ndence with the set of idemp otents of the ring Ξ( G ). By a standard argument f r om commutativ e ring theory , this set is in one to one corresp ondence with t he set of connected comp o nents of the sp ectrum of Ξ( G ). It fo llows that if G has o dd order, the spectrum of G is connected if and only if Ξ( G ) × = {± 1 } . By Theorem 7.9, this is equiv alen t to saying that G is solv able. The following theorem is an ana logue of Y oshida’s characterization ( [10]) of the unit group of the usual Burnside ring : 8.4. Theorem : L et G b e a finite gr oup, and let m = ( m T ,S ) ( T ,S ) ∈ Π( G ) b e a se quenc e of inte gers in {± 1 } in d exe d by Π( G ) , c onstant on G - c onjugacy classes of slic es. Then the se quenc e [ m ] = ( m T ,S ) ( T ,S ) ∈ [Π( G )] of r epr esentatives lies in the image of the r estricte d ghost map Φ × if and only if f o r any ( T , S ) ∈ Π( G ) , the map g ∈ N G ( T , S ) /S 7→ m , /m T ,S ∈ {± 1 } is a gr oup homom orphism. Pro of : Let X f → Y b e a morphism of finite G -sets. It follows from Lemma 4.3 tha t f or any ( T , S ) ∈ Π( G ), the monoid of endomorphisms of G/S → G/T in the category G - Mor is a ctually a gro up, isomorphic to N G ( T , S ) /S . This gr oup acts on the set of morphisms from G/S → G/T to X f → Y , by pre- comp osition : if G/S / / α   G/T β   X f / / Y is a morphism in G - M o r , a nd if g ∈ N G ( T , S ), t he morphism g ( α, β ) = ( g α, g β ) is defined b y ( g α )( xS ) = xg S and ( g β )( xT ) = xg T , for any g ∈ T . 23 The mor phism ( α , β ) is in v arian t under g ∈ N G ( T , S ) if a nd only if α ( g S ) = α ( S ), i.e. if ( α , β ) factors as G/S / /   G/g T   G/ / / α   G/ β   X f / / Y It follow s that the n umber of fixed p oin ts of g ∈ N G ( T , S ) /S on the set of homomo r phisms from G/S → G/ T to X f → Y , i.e. the v alue at g S of the corresp onding p erm utation c har a cter θ T ,S of N G ( T , S ) /S , is equal to φ , ( X f → Y ). This sho ws more g enerally that for any u ∈ Ξ( G ), the corresp ondence θ T ,S : g ∈ N G ( T , S ) /S 7→ φ , ( u ) is a generalized c haracter of the group N G ( T , S ) /S . No w if u ∈ Ξ( G ), t his generalized character has all its v alues in {± 1 } . It follo ws that h θ T ,S , θ T ,S i G = 1, hence θ T ,S is up to a sign equal to an irreducible c ha racter of N G ( T , S ) /S , of degree 1. Hence θ T ,S /θ T ,S (1) is a group homomorphism from N G ( T , S ) /S to {± 1 } . Thus if m ∈ Im (Φ × ), then the map g ∈ N G ( T , S ) /S 7→ m , /m T ,S ∈ {± 1 } is a group homomorphism. Con vers ely , if m = ( m T ,S ) ( T ,S ) ∈ Π( G ) is a G - inv ar ia n t sequenc e with v alues in {± 1 } , suc h tha t for any ( T , S ) ∈ Π( G ), the map g ∈ N G ( T , S ) /S 7→ m , /m T ,S ∈ {± 1 } is a group homomorphism θ T ,S , then X g ∈ N G ( T ,S ) /S m , = m T ,S | N G ( T , S ) /S | h θ T ,S , 1 i G is an in teger m ultiple of | N G ( T , S ) /S | . By Theorem 6.1, it fo llo ws that [ m ] = Φ( u ), for some u ∈ Ξ( G ). Then u ∈ Ξ( G ) × , by Lemma 8.2, and this completes the pro of. 8.5. In the case of the usual Burnside ring B ( G ), the corresp ondence sending a finite group G to the unit group B ( G ) × can be endow ed with a structure of biset functor, using tens or induction (see e.g. [4], Prop osition 5.5). One may ask whether a similar structure exists fo r the g roup of units of the section Burnside ring : 24 8.6. Prop osition : The c orr esp ondenc e sending a finite g r oup G to Ξ( G ) × c annot b e endowe d with a structur e of b i s e t f unctor. Pro of : If suc h a structure exists, o ne ma y view it as a biset functor Ξ × with v alues in F 2 -v ector spaces. Ob viously Ξ × ( 1 ) = F 2 . This shows that there are t w o subfunctors F 2 ⊂ F 1 of Ξ × suc h that F 1 /F 2 is isomorphic to the simple functor S 1 , F 2 . An easy computation (a sp ecial case of Theorem 16.13 in t he App end ix) sho ws moreo ver that Ξ × ( C 2 ) ∼ = ( F 2 ) 3 , where C 2 is a group of order 2. As S 1 , F 2 ( C 2 ) ∼ = ( F 2 ) 2 (see e.g. [5] Prop osition 4.4.6), this shows that either ( F / F 1 )( C 2 ) ∼ = F 2 and F 2 ( C 2 ) = { 0 } , or F ( C 2 ) = F 1 ( C 2 ) and F 2 ( C 2 ) ∼ = F 2 . In the first case, there are subfunctors F 3 and F 4 of F with F 1 ⊆ F 4 ⊂ F 3 ⊆ F and F 3 /F 4 ∼ = S C 2 , F 2 . In t he la t ter case, there are subfunctors F 4 ⊂ F 3 of F 2 suc h that F 3 /F 4 ∼ = S C 2 , F 2 . In any case dim F 2 Ξ ×  ( C 2 ) 2  ≥ dim F 2 S 1 , F 2  ( C 2 ) 2  + dim F 2 S C 2 , F 2  ( C 2 ) 2  . No w dim F 2 S 1 , F 2  ( C 2 ) 2  = 4 b y [5] Prop osition 4 .4 .6 or Corollar y 10.5.6, and dim F 2 S C 2 , F 2  ( C 2 ) 2  = 5 b y insp e ction, using [5] Prop osition 4.4.6. Th us dim F 2 Ξ ×  ( C 2 ) 2  ≥ 9. When G is a finite ab elian group, the unit g roup Ξ( G ) × can b e determined explicitly (see Theorem 16 .13). In pa rticular ( 8 . 7 ) Ξ  ( C 2 ) 2  × ∼ = ( F 2 ) 7 . This con tradiction sho ws that the biset functor Ξ × cannot exist. 8.8. Remark : Boltje and Pfeiffe r ([1]) ha ve describ ed an efficien t algorithm to compute t he unit group of the ordinary Burnside ring. A straigh tforw ard adaptation of this algo rithm to the ring Ξ( G ) allows for a quic k computation of the gr oup Ξ( G ) × , for not to o lar g e finite groups G , using GAP4 ([7]). These computations agree in particular with 8 .7. 25 I I - The se c tion B urnside ring 9. Galois morphisms of G -se ts 9.1. Definition : L et G b e a gr oup. A morphi s m f : X → Y of G -sets is a G a lois morphism if for any x, x ′ ∈ X such that f ( x ) = f ( x ′ ) , ther e exists ϕ ∈ Aut G ( X ) such that f ◦ ϕ = f a nd ϕ ( x ) = x ′ . 9.2. E xample : An y injectiv e morphism of G -sets is a Galois morphism, for trivial reasons. 9.3. P rop osition : L et f : X → Y b e a morphism of G -sets. The fol lowing c onditions ar e e quivalent: 1. f i s a Galois morphism. 2. ∀ x, x ′ ∈ X , f ( x ) = f ( x ′ ) = ⇒ G x = G x ′ . Pro of : Supp ose first that f is a Galois morphism. Then if f ( x ) = f ( x ′ ), there exists a G -auto mo r phism ϕ of X suc h that ϕ ( x ) = x ′ . This implies G x ≤ G x ′ , hence G x = G x ′ b y symmetry . Th us Condition 1 implies Condi- tion 2. Con vers ely , supp ose that Condition 2 holds, and let x, x ′ ∈ X with f ( x ) = f ( x ′ ). There are t w o cases : • Either x and x ′ are in the same G -orbit ω . Let Z = X − ω . Define ϕ : X → X by ϕ ( t ) = t if t ∈ Z and ϕ ( ux ) = u x ′ if u ∈ G . This is a w ell defined G -automorphism of X , since G x = G x ′ . Clearly ϕ ( x ) = x ′ , and f ◦ ϕ ( t ) = f ( t ) if t ∈ Z . Moreov er f ◦ ϕ ( u x ) = f ( ux ′ ) = uf ( x ′ ) = uf ( x ) = f ( ux ) for an y u ∈ G . Hence f ◦ ϕ = f . • If x and x ′ are in differen t G - orbits ω and ω ′ , resp ectiv ely , let Z = X − ( ω ⊔ ω ′ ). Define ϕ : X → X b y ϕ ( t ) = t if t ∈ Z , and ϕ ( u x ) = ux ′ and ϕ ( ux ′ ) = ux , for any u ∈ G . This is a w ell defined G -auto mo r phism of X , since G x = G x ′ . Clearly ϕ ( x ) = x ′ , and f ◦ ϕ ( t ) = f ( t ) if t ∈ Z . Moreo v er f ◦ ϕ ( u x ) = f ( ux ′ ) = uf ( x ′ ) = uf ( x ) = f ( ux ) 26 for an y u ∈ G . Also f ◦ ϕ ( ux ′ ) = f ( ux ) = uf ( x ) = uf ( x ′ ) = f ( ux ′ ) , for an y u ∈ G . Hence f ◦ ϕ = f . It follows that f is a Galo is morphism, and Condition 2 implies Conditio n 1. 9.4. Corollary : L et f : X → Y b e a morphism o f G -sets. Then f is a Galois morph i s m if and only if for any y ∈ f ( X ) , ther e exists a norm al sub gr oup N y of G y such that G x = N y for any x ∈ f − 1 ( y ) . Pro of : Supp ose first that f is a Galois morphism. If x ∈ X , set y = f ( x ), and c ho ose g ∈ G y . Then f ( g x ) = g y = y = f ( x ), th us G g x = g G x = G x , thus G x E G y . Moreov er G x do es not dep end on x ∈ f − 1 ( y ), b y Prop osition 9.3. Con vers ely , supp ose that for an y y ∈ f ( X ), there exists N y E G y suc h that G x = N y for any x ∈ f − 1 ( y ). Then obv iously G x = G x ′ = N y if f ( x ) = f ( x ′ ) = y , so f is a Galois morphism, by Prop osition 9.3. 9.5. Remark : In particular, when ( T , S ) is a slice of G , t he pro jection morphism G/S → G/T is a Galo is morphism o f G -sets if and only if S E T , i.e. if ( T , S ) is a se ction of G . 9.6. Lemm a : L et f : X → Y b e a mo rp hism of G - s e ts, and j : Y ֒ → Z b e an in je ctive morphis m of G -sets. Then f is a Galois morphi s m if and only if j ◦ f is a Galois morphism. Pro of : This is straightforw ard. 9.7. Lemma : L et X a / / b   Y c   Z d / / T b e a c artesian s q uar e of G -se ts. I f c is a Galo i s morphism, then b is a Galois morphism. Pro of : Supp ose that x, x ′ ∈ X a re suc h that b ( x ) = b ( x ′ ). Then ca ( x ) = db ( x ) = db ( x ′ ) = ca ( x ′ ) . 27 If c is a Galois morphism, it follows that G a ( x ) = G a ( x ′ ) . Let g ∈ G x . Then g ∈ G a ( x ) = G a ( x ′ ) , and g ∈ G b ( x ) = G b ( x ′ ) , th us b ( x ′ ) = g b ( x ′ ) = b ( g x ′ ) a ( x ′ ) = g a ( x ′ ) = a ( g x ′ ) . It follo ws that x ′ = g x ′ , so G x ≤ G x ′ , and G x = G x ′ b y symmetry . Hence f is a Galois morphism, by Prop osition 9.3 . 9.8. R emark : In particular, any morphism isomorphic to a Galois mor- phism (that is, when a and d are isomorphisms) is a Galo is morphism. 9.9. Corollary : L et f : X → Y b e a Galois morphism of G -sets. If X 1 is a G -subset of X , the r estricte d mo rphism f | X 1 : X 1 → f ( X 1 ) is a Galois morphism of G -sets. Pro of : This follows from Lemma 9.7 and Lemma 9.6, since the square X 1 i / / f 1   X f   f ( X 1 ) j / / Y is cartesian, where i and j are the injection maps, and f 1 = f | X 1 . 9.10. Pr op osition : A morphism f : X → Y 1 ⊔ Y 2 is a Galois morphism if and only if the r estricte d morphisms f − 1 ( Y 1 ) → Y 1 and f − 1 ( Y 2 ) → Y 2 ar e Galois morphisms. Pro of : Set X i = f − 1 ( Y i ), and denote by f i : X i → Y i the restriction of f , for i = 1 , 2. Assume first that f is a Galois morphism. If x, x ′ ∈ X 1 ha v e the same ima g e under f 1 , then they ha ve t he same image under f , and G x = G x ′ . So f 1 is a Ga lois morphism, a nd f 2 is also a Galois morphism, b y symmetry . Con vers ely , suppo se that f 1 and f 2 are Galois morphisms. If x, x ′ ∈ X are suc h that f ( x ) = f ( x ′ ) ∈ Y 1 , then x, x ′ ∈ X 1 , and f 1 ( x ) = f 1 ( x ′ ). Th us G x = G x ′ , as f 1 is a Galois morphism. If f ( x ) = f ( x ′ ) ∈ Y 2 , the argumen t is similar, with f 1 replaced b y f 2 . In any case G x = G x ′ , and f is a Galois morphism, b y Prop o sition 9.3. 9.11. Prop osition : L et G and H b e finite gr o ups , and U b e an ( H , G ) - 28 biset. If f : X → Y is a Galois morphism of G -sets, then U × G f : U × G X → U × G Y and U ◦ G f : U ◦ G X → U ◦ G Y ar e Galois mo rp h isms o f H -sets. Pro of : F or U × G f , let ( u, G x ) and ( u ′ , G x ′ ) b e elemen ts of U × G X ha ving the same imag e under U × G f . This means that  u, G f ( x )  =  u ′ , G f ( x ′ )  , i.e. that there exist g ∈ G suc h that ug − 1 = u ′ and g f ( x ) = x ′ . Thus f ( g x ) = f ( x ′ ), and G g x = G x ′ , since f is a Galois morphism. No w if h ∈ H stabilizes ( u, G x ), i.e. if ( hu, G x ) = ( u, G x ), there exists a ∈ G suc h that hu = ua and a − 1 x = x . Hence h ( u ′ , G x ′ ) = ( hu ′ , G x ′ ) = ( hug − 1 , G x ′ ) = ( uag − 1 , G x ′ ) = ( u, G ag − 1 x ′ ) . Since a ∈ G x , it follo ws t ha t g a ∈ G g x = G x ′ , so ag − 1 x ′ = g − 1 x ′ , and h ( u ′ , G x ′ ) = ( u, G g − 1 x ′ ) = ( ug − 1 , G x ′ ) = ( u ′ , G x ′ ) . This shows that h stabilizes ( u ′ , G x ′ ). By symmetry , the stabilizers of ( u, G x ) and ( u ′ , G x ′ ) in H a r e equal, a nd U × G f is a G alois morphism of H - sets. No w recall from [2] that U ◦ G X is the H - subset of U × G X defined by U ◦ G X = { ( u, G x ) ∈ U × G X | ∀ g ∈ G, ug = g = ⇒ g x = x } , and that t he map U ◦ G f is the res triction of U × G f to U ◦ G X . By Lemma 9.6 and Coro llary 9.9, the morphism U ◦ G f is also a Galois morphism of H -sets. 9.12. Corollary : L et f : X → Y b e a Galois m orphism of G -sets. If H is a sub gr oup of G , the r estriction Res G H f : Res G H X → R es G H Y is a Galois morphism of H -sets. Pro of : In Prop osition 9.11, set U = G , view ed as and ( H , G )-biset for left and righ t m ultiplication. Let G and H b e groups. A morphism f : U → U ′ of ( H , G )-bisets is called a Galois morphism if it is a Galois morphism of ( H × G op )-sets. Then : 9.13. Prop osition : L et G , H and K b e gr oups. I f f : U → U ′ is a Galois morphism of ( H , G ) - bisets, and g : V → V ′ is a Galois morphism o f ( K , H ) -bisets, then g × H f : V × H U → V ′ × H U ′ is a Galois morphism of ( K , G ) -bis e ts. 29 Pro of : Let ( v , H u ) and ( v ′ , H u ′ ) b e elemen ts o f V × H U with the same image under g × H f . It means that there exists h ∈ H suc h that g ( v ′ ) = g ( v ) h = g ( v h ) and f ( u ′ ) = h − 1 f ( u ) = f ( h − 1 u ) . As g is a Galo is morphism, the stabilizers of v ′ and v h in K × H op are equal. Similarly , as f is a Galois morphism, the stabilizers of u ′ and h − 1 u in H × G op are equal. No w let ( k , g ) ∈ K × G op suc h t ha t k ( v ′ , H u ′ ) g − 1 = ( v ′ , H u ′ ). It means that there exists a ∈ H suc h that k v ′ a − 1 = v ′ and au ′ g − 1 = u ′ . Hence k v ha − 1 = v h and ah − 1 ug − 1 = h − 1 u . It follow s that k ( v , H u ) g − 1 = ( k v , H ug − 1 ) = ( v ha h − 1 , H ug − 1 ) = ( v , H hah − 1 ug − 1 ) = ( v , H hh − 1 u ) = ( v , H u ) . By symmetry , the stabilizers o f ( v , H u ) and ( v ′ , H u ′ ) in K × G op are equal. Th us ( g × H f ) is a Galois morphism of ( K , G )-bisets . 9.14. Corollary : L et G and H b e gr oups. I f f : X → Y is a Galois morphism of G sets and g : Z → T is a Galois morphism of H -sets, then f × g : X × Z → Y × T is a Galois morph ism of ( G × H ) -sets. Pro of : Consider f as a morphism of ( G, 1 )-bisets, and g as a morphism of ( 1 , H op )-bisets. 9.15. N otation : L et G - Mo r Gal denote the ful l sub c ate gory of G - Mor c onsisting of Galois morphi s m s of G -sets. 9.16. Notation : L et X f → Y b e a morphism of G -sets. F or x ∈ X , set G f x = . L et ∼ f b e the r elation on X define d by x ∼ f x ′ ⇔ ∃ g ∈ G f x , g x = x ′ . 30 9.17. Lemma : With this notation : 1. The r elation ∼ f is an e quivalenc e r elation o n X . L et X Gal f denote the set of e quivalenc e clas ses, and let γ X,f : X → X Gal f denote the pr oje ction map. 2. If x, x ′ ∈ X and x ∼ f x ′ , then g x ∼ f g x ′ for an y g ∈ G . Henc e ther e exists a unique structur e o f G -set on X Gal f such that γ X,f is a morp h ism of G -se ts. 3. Ther e is a unique map f Gal : X Gal f → Y such that the diagr am X f / / γ X,f   Y X Gal f f Gal = = { { { { { { { { is c om mutative. Pro of : F or Assertion 1, the relation ∼ f is clearly reflexiv e, since G x ≤ G f x . No w if x, x ′ ∈ X and x ∼ f x ′ , let g ∈ G f x suc h that g x = x ′ . There exist r ∈ N , elemen ts z 1 , . . . , z r in X and elemen ts g 1 , . . . , g r of G suc h that g i z i = z i and f ( z i ) = f ( x ), for i = 1 , . . . , r , and suc h that g = g 1 · · · g r . It follo ws that f ( x ′ ) = g 1 · · · g r f ( x ) = g 1 · · · g r f ( z r ) = g 1 · · · g r − 1 f ( g r z r ) = g 1 · · · g r − 1 f ( z r ) = g 1 · · · g r − 1 f ( x ) , th us f ( x ′ ) = f ( x ), b y induction on r . It follo ws that G f x ′ = G f x , hence that x ′ ∼ f x , since x = g − 1 x ′ . So the relation ∼ f is symmetric. Finally ∼ f is transitiv e : if x, x ′ , x ′′ ∈ X , if x ∼ f x ′ and x ′ ∼ f x ′′ , there exist g ∈ G f x and g ′ ∈ G f x ′ suc h that g x = x ′ and g ′ x ′ = x ′′ . But the previous argumen t sho ws that G f x = G f x ′ = G f x ′′ . Hence g ′ g ∈ G f x , and x ∼ f x ′′ , since g ′ g x = x ′′ . Assertion 2 follo ws from the straightforw ard fact that g ( G f x ) = G f g x for an y x ∈ X and an y g ∈ G . Assertion 3 follow s as we ll, since x ∼ f x ′ implies f ( x ) = f ( x ′ ). 9.18. Prop osition : L et G b e a finite gr o up. 1. L et X f → Y b e a morphism of G -sets. Then the morphism X Gal f f Gal − → Y is a Galois morphi s m of G -sets. 31 2. If X f / / α   Y β   A a / / B is a morphis m in the c ate gory G - M o r , and if A a → B is a Galois morphism of G -sets, then ther e exists a unique morphism of G -sets e α : X Gal f → A such that the diagr am X f / / γ X,f   α Y X Gal f e α   f Gal / / Y β   A a / / B is c om mutative. 3. The c orr es p ondenc e sending X f → Y to X Gal f f Gal − → Y is a functor fr om G - Mor t o G - Mor Gal , and this functor is left adjo int to the f o r getful functor G - Mo r Gal → G - Mo r . Pro of : When x ∈ X , let e x = γ X,f ( x ) ∈ X Gal f denote its equiv alence class for the relation ∼ f . Let g ∈ G . Then g e x = e x if and only if g x ∼ f x , i.e. if there exists h ∈ G f x suc h that g x = hx , or equiv alen tly h − 1 g ∈ G x . Hence the stabilizer of e x in G is equal to G f x · G x = G f x , since G x ≤ G f x . So if x, x ′ ∈ X are suc h that f Gal ( e x ) = f Gal ( e x ′ ), i.e. if f ( x ) = f ( x ′ ), t hen the stabilizers G f x of e x a nd G f x ′ of e x ′ are equal, since G f x dep ends only on f ( x ). Assertion 1 follo ws, b y Prop osition 9 .3 . Let x, x ′ ∈ X suc h that x ∼ f x ′ . Then there exist r ∈ N , elemen ts z 1 , . . . , z r in X and elemen ts g 1 , . . . , g r of G suc h that g i z i = z i and f ( z i ) = f ( x ), fo r i = 1 , . . . , r , and suc h that g = g 1 · · · g r . It follo ws that β f ( z i ) = aα ( z i ) = β f ( x ) = aα ( x ) , for i = 1 , . . . , r . By Prop osition 9.3, since A a → B is a Galois morphism, this implies G α ( z i ) = G α ( x ) . Moreo v er G z ≤ G α ( z ) for an y z ∈ X . Th us g i ∈ G α ( z i ) = G α ( x ) for i = 1 , . . . , r . It follows that g = g 1 · · · g r ∈ G α ( x ) , hence α ( x ′ ) = g α ( x ) = α ( x ). This shows the existence of a map e α : X Gal f → A , sending the equiv alence class of x ∈ X for ∼ f to α ( x ). Such a map is ob viously unique, and it is a mor phism o f G -sets. This pro v es Assertion 2. 32 F or Assertion 3, supp ose that X f / / α   Y β   X ′ f ′ / / Y ′ is a morphism fro m X f → Y to X ′ f ′ → Y ′ in the category G - Mor . Then one can comp ose this morphism with the morphism X ′ f ′ / / γ X ′ ,f ′   Y ′ X ′ Gal f ′ f ′ Gal / / Y ′ . This yields a morphism from X f → Y to X ′ Gal f ′ f ′ Gal − → Y ′ , whic h is a Galois morphism by Assertion 1 . By Assertion 2, this comp osition factors in a unique w ay thro ug h the morphism X Gal f f Gal − → Y . In o ther w ords there is a unique morphism of G -sets e α : X Gal f → X ′ Gal f suc h that the dia g ram X f / / α ) ) R R R R R R R R R R R R R R R R R γ X,f   Y β ) ) R R R R R R R R R R R R R R R R R R X ′ f ′ / / γ X ′ ,f ′   Y ′ X Gal f f Gal / / e α ' ' P P P P P P P P P P P P P P P Y β ( ( P P P P P P P P P P P P P P P P P P X ′ Gal f ′ f ′ Gal / / Y ′ is comm utativ e. Clearly , the map ( α , β ) 7→ ( e α, β ) endow s the corresp ondence  X f → Y  7→  X Gal f f Gal − → Y  with a structure of functor f rom G - M o r to G - Mor Gal . Moreo ver, for an y Galois morphism A a → B , Assertion 2 yields a bijection Hom G - Mo r  X f → Y , A a → B  ∼ = Hom G - Mo r Gal  X Gal f f Gal − → Y , A a → B  , 33 and this bijection is clearly functorial with resp ect to X f → Y and A a → B . Assertion 3 follows. 9.19. E xample : Let ( T , S ) b e a slice of G , and f b e the pro jection map from X = G/S to Y = G/T . Then for x = S ∈ X , the g roup G f x is the group generated by the stabilizers G tS , for t ∈ T . As G tS = t S , it follows that G f x is equal to the n o rm al closur e S E T of S in T . In this case moreov er, the G -set X Gal f is isomorphic to G/ ( S E T ), and the map f Gal is the pro jection to G/T . 10. The section Burns ide fun ctor 10.1. P rop osition : If X f → Y and X ′ f ′ → Y ′ ar e Galois morphism s of G -sets, then X ⊔ X ′ f ⊔ f ′ − → Y ⊔ Y ′ and X × X ′ f × f ′ − → Y × Y ′ ar e Galois morphism s of G -sets. Pro of : The case of disjoin t union follo ws from Prop osition 9.10 and Corol- lary 9.9. Moreov er Pro p osition 9.1 4 sho ws that if f : X → Y a nd f ′ : X ′ → Y ′ are Galois morphisms of G - sets, then f × f ′ : X × X ′ → Y × Y ′ is a G alois morphism of ( G × G )- sets. By Corollary 9.12, the restriction of this morphism t o the diag onal G ≤ G × G is a Galois morphism of G -sets. 10.2. Definition : L et G b e a finite gr oup. The section Burnside group Γ( G ) of G is the sub gr oup of the slic e Burnside ring Ξ( G ) gener ate d by the classes of Galois morphis m s of G -sets. By Pr op osition 10.1, the gr oup Γ( G ) is actual ly a subring of Ξ( G ) , c al le d the se ction Burnsid e ring of G . 10.3. Lemma : L et f : X → Y b e a Galois morphism of finite G -sets. Then in the gr oup Γ( G ) π ( X f → Y ) = X x ∈ [ G \ X ] h G f ( x ) , G x i G = X y ∈ [ G \ Y ] | G y \ f − 1 ( y ) | h G y , G x y i G , 34 wher e x y is chosen in f − 1 ( y ) , for e ac h y ∈ [ G \ f ( X )] . Pro of : The first formula fo llows from Lemma 3.4. F or the second one, write π ( X f → Y ) = X y ∈ [ G \ Y ] X x ∈ [ G y \ f − 1 ( Y )] π ( G/G x → G/G y ) , and observ e that G x = G x ′ if f ( x ) = f ( x ′ ). 10.4. Corollary : The elements h T , S i G , wher e ( T , S ) runs thr ough a set [Σ( G )] of r epr esentatives of c onjugacy classes of se ctions of G , form a b asis of Γ( G ) . Pro of : Indee d, these elemen ts g enerate Γ( G ), by Prop o sition 10.3, and they are linearly indep enden t, by 4.6. 10.5. Remark : This also shows that Γ ( G ) is the quotien t of the free ab elian group o n the set of isomorphism classes [ X f − → Y ] of Galois morphisms of finite G -sets, b y the subgroup generated by elemen ts of the f orm [( X 1 ⊔ X 2 ) f 1 ⊔ f 2 − → Y ] − [ X 1 f 1 → f ( X 1 )] − [ X 2 f 2 → f ( X 2 )] , whenev er X f → Y is a Galois morphism of finite G -sets with a decomposition X = X 1 ⊔ X 2 as a disjoin t union of G -sets, where f 1 = f | X 1 and f 2 = f | X 2 . 10.6. Theorem : 1. L et G and H b e finite gr oups, and let U b e a finite ( H , G ) -biset. The functor ( X f → Y ) 7→ ( U × G X U × G f − → U × G Y ) fr om G - Mo r Gal to H - Mor Gal induc es a gr o up homomorphism Γ( U ) : Γ( G ) → Γ( H ) . 2. The c orr esp ondenc e G 7→ Γ( G ) is a Gr e en biset functor. Pro of : This follows from Theorem 3.9 a nd Prop osition 9.11 . 10.7. R emark : It follows fro m Remark 9.2 tha t the imag e of the morphism i : B → Ξ of Prop osition 3.11 is actually con tained in Γ. Th us i is a morphism of Green biset functors from B to Γ. 35 11. Se ctions and ghost map 11.1. Notat ion : If ( T , S ) is a sli c e of G , denote by ψ T ,S : Γ( G ) → Z the r estriction of the ring homomo rphism φ T ,S : Ξ( G ) → Z . 11.2. Lemma : L et ( T , S ) , ( T ′ , S ′ ) ∈ Π( G ) . Then ψ T ,S = ψ T ′ ,S ′ if and only if the se ctions ( T , S E T ) and ( T ′ , S ′ E T ′ ) ar e c onjuga te in G (r e c a l l that S E T denotes the norm a l closur e of S in T ) . In p articular ψ T ,S = ψ T ,S E T . Pro of : This follow s from Prop osition 9.18 and Remark 9.19, but the fol- lo wing is a short direct pro of : by Lemma 4.3, ψ T ,S = ψ T ′ ,S ′ if and only if fo r an y section ( V , U ) of G |{ g ∈ G/U | ( T , S ) g  ( V , U ) }| = |{ g ∈ G/U | ( T ′ , S ′ ) g  ( V , U ) }| . T aking ( V , U ) = ( T , S E T ) sho ws that there exists g ∈ G suc h that ( T ′ , S ′ ) g ≤ ( T , S E T ). This implies ( T ′ , S ′ E T ′ ) g  ( T , S E T ). T aking now ( V , U ) = ( T ′ , S ′ E T ′ ) sho ws that ( T ′ , S ′ E T ′ ) g ′  ( T , S E T ), fo r some g ′ ∈ G . It follows that ( T ′ , S ′ E T ′ ) g = ( T , S E T ). 11.3. Theorem : The m a p (c a l le d the ghost map for Γ( G ) ) Ψ = Y ( T ,S ) ∈ [Σ( G )] ψ T ,S : Γ( G ) → Y ( T ,S ) ∈ [Σ( G )] Z is an i n je ctive ring homomorphism , with finite c okern e l a s morphism of ab elian gr oups. Pro of : The pro of is exactly the same as for Theorem 4 .6 : b y Lemma 3.5, the elemen ts h T , S i G , for ( T , S ) ∈ [Σ( G )], generate Γ( G ). Supp ose that there is a non zero linear com bination in the kernel of Ψ Λ = X ( T ,S ) ∈ [Σ( G )] λ T ,S h T , S i G with integer co efficien ts λ T ,S ∈ Z , f or ( T , S ) ∈ [Σ( G )]. Extend λ to a function Σ( G ) → Z , constan t on conjug acy class es. Let ( Y , X ) b e an eleme n t of Σ( G ), maximal for the relation  , such that λ Y , X 6 = 0. Then since b y Lemma 4.3 ψ Y , X ( G/S → G/T ) = |{ g ∈ G/S | ( Y g , X g )  ( T , S ) }| , 36 it follow s that ψ Y , X (Λ) = X ( T ,S ) ∈ [Σ( G )] λ T ,S ψ Y , X ( G/S → G/T ) = λ Y , X ψ Y , X ( G/X → G/ Y ) = λ Y , X | N G ( X , Y ) / X | = 0 . Hence λ Y , X = 0 , and this contradiction sho ws tha t Ψ is injectiv e (a nd in particular, we recov er the fact that the elemen ts h T , S i G , for ( T , S ) ∈ [Σ( G )], form a Z -basis of Γ ( G )). Th us Ψ is an injectiv e mo r phism b etw een free ab elian groups with the same finite ra nk, hence it ha s finite cok ernel. 11.4. Corollary : Set Q Γ( G ) = Q ⊗ Z Γ( G ) , a n d Q Ψ = Q ⊗ Z Ψ . Then Q Ψ : Q Γ( G ) → Y ( T ,S ) ∈ [Σ( G )] Q is an isom orphism of Q -algebr as. 12. Se ctions and idemp oten ts Corollary 11.4 sho ws that Q Γ( G ) is a split semisimp le comm utativ e algebra. Its primitiv e idemp otents are indexed b y sections of G , up to conjugation : they are the in vers e imag es under Q Ψ of the primitive idemp otents of the algebra Q ( T ,S ) ∈ [Σ( G )] Q : 12.1. Notation : If ( T , S ) ∈ Σ( G ) , deno te by γ G T ,S the unique element of Q ⊗ Z Γ( G ) such that ∀ ( Y , X ) ∈ Σ( G ) , Q ψ Y , X ( γ G T ,S ) =  1 if ( Y , X ) = G ( T , S ) 0 otherwi s e The elements γ G T ,S , for ( T , S ) ∈ Σ( G ) , ar e the primitive idemp otents of Q Γ( G ) . 37 12.2. Theorem : L et  denote the r estriction of the r elation  to Σ( G ) . Then for ( T , S ) ∈ Σ( G ) γ G T ,S = 1 | N G ( T , S ) | X ( V ,U )  ( T ,S ) | U | µ Σ  ( V , U ) , ( T , S )  h V , U i G , wher e µ Σ is the M¨ obius function of the p oset (Σ( G ) ,  ) . Pro of : The pro of is the same as the pro of of Theorem 5.2. 12.3. P rop osition : L et ( X , ≤ ) b e a finite p os e t. L et ϕ : X → X b e a map of p o s ets such that ϕ ◦ ϕ = ϕ and x ≤ ϕ ( x ) f o r any x ∈ X . Then the M¨ obius function µ Y of the subp oset Y = ϕ ( X ) of X i s giv e n by ( 12 . 4 ) ∀ y , z ∈ Y , µ Y ( y , z ) = X y ≤ u ∈X ϕ ( u )= z µ X ( y , u ) , wher e µ X is the M¨ obius function of ( X , ≤ ) . Pro of : F or y , z ∈ Y , denote b y m ( y , z ) the righ t hand side of Equation 12.4. Then for y , t ∈ Y X z ∈Y y ≤ z ≤ t m ( y , z ) = X z ∈Y y ≤ z ≤ t X y ≤ u ∈X ϕ ( u )= z µ X ( y , u ) = X ( z ,u ) ∈P µ X ( y , u ) , where P = { ( z , u ) | z ∈ Y , u ∈ X , y ≤ u ≤ ϕ ( u ) = z ≤ t } . Set Q = { u ∈ X | y ≤ u ≤ t } . If ( z , u ) ∈ P , then clearly u ∈ Q . Conv ersely , if u ∈ Q , then  ϕ ( u ) , u  ∈ P : indeed ϕ ( u ) ∈ Y = ϕ ( X ), and moreo ver ϕ ( u ) ≤ ϕ ( t ) = t , since t ∈ ϕ ( X ) and ϕ ◦ ϕ = ϕ . Now the maps ( z , u ) ∈ P 7→ u ∈ Q and u ∈ Q 7→  ϕ ( u ) , u  ∈ P are mutual in v erse bijections. It follows that X z ∈Y y ≤ z ≤ t m ( y , z ) = X u ∈X y ≤ u ≤ t µ X ( y , u ) . This is equal to 1 if y = t , and to zero otherwise. The prop osition follo ws. 38 12.5. Corollary : L et ( V , U ) and ( T , S ) b e se ctions of G . Then ( 12 . 6 ) µ Σ( G )  ( V , U ) , ( T , S )  = µ ( V , T )  X U ≤ X ≤ V X E T = S µ ( U, X )  . In p articular in Q Γ( G ) γ G T ,S = 1 | N G ( T , S ) | X U E V ≤ T U ≤ X ≤ V X E T = S | U | µ ( U, X ) µ ( V , T ) h V , U i G . Pro of : F or Equation 12.6, apply Prop osition 12.3 to the p oset X =  Π( G ) ,   , and to the map ϕ : X → X defined b y ϕ  ( Y , X )  = ( Y , X E Y ) , and then use Equation 5.6. Then substitute the v alue of µ Σ  ( V , U ) , ( T , S )  in the form ula of Theorem 12.2. 13. The image o f the g h ost map The follo wing is a c haracterization of the image of the ghost map for t he section Burnside ring, similar to Theorem 6 .1 : 13.1. Theorem : L et G b e a finite gr oup. L et m =  m ( T , S )  ( T ,S ) ∈ Σ( G ) b e a se q uen c e of inte gers indexe d b y Σ( G ) , c o n stant on G -c onjugacy classes of se ctions. Then the se quen c e [ m ] =  m ( T , S )  ( T ,S ) ∈ [Σ( G )] of r epr esentatives lies in the image of the ghost ma p Ψ if and only i f , fo r any se ction ( T , S ) of G X g ∈ N G ( T ,S ) /S m  , E  ≡ 0  mo d . | N G ( T , S ) /S |  . Pro of : The pro of is v ery similar to the pro of of Theorem 6.1, with tw o differences : the first one is that the p oset Π( G ) of slices of G has to b e replaced b y the p o set Σ( G ) of sections of G . The second one is that the slice ( < g T >, ) has to b e c ha ng ed to the corr espo nding section 39  , E gT  of G . Apart from these t w o differences, the pro of go es through without c hanges. 14. Prime sp ectr u m 14.1. Notation : L et p den ote either 0 or a pri m e numb e r. • If ( T , S ) ∈ Σ( G ) , denote by J T ,S,p the p rime ide al I T ,S,p ∩ Γ( G ) of Γ( G ) . In other wo r d s J T ,S,p is the k ernel o f the ring homomorphism Γ( G ) ψ T ,S − → Z → Z /p Z , wher e the right hand side map is the pr oje ction. • L et Ω( G ) d enote the set of triples ( T , S, p ) , wher e ( T , S ) ∈ Σ( G ) is such that | N G ( T , S ) /S | 6≡ 0 (mo d . p ) . The group G acts on Ω( G ), b y g ( T , S, p ) = ( g T , g S , p ), for g ∈ G , and the ideal J T ,S,p only dep ends on the G -o rbit o f ( T , S, p ). Conv ersely : 14.2. P rop osition : L et I b e a p rime id e al of Γ( G ) , and R = Γ( G ) / I . Denote by ψ : Γ ( G ) → R the pr oje ction map, and d e note by p ≥ 0 the char acteristic of R . Then R ∼ = Z /p Z a nd : 1. If p = 0 , ther e exists a se ction ( T , S ) of G such that ψ = ψ T ,S , and ( T , S ) is unique up to G -c onj uga tion , with this pr o p erty. 2. If p > 0 , ther e exists a se ction ( T , S ) of G such that ψ is the r e duction mo dulo p of ψ T ,S and N G ( T , S ) /S is a p ′ -gr oup, and ( T , S ) i s unique up to G -c o n jugation, wi th these pr op erties. In p articular, ther e exists a unique ( T , S, p ) ∈ Ω( G ) , up to c onjuga tion , such that I = J T ,S,p . Pro of : The pro of is exactly the same a s the pro of of Prop osition 7.2, with slices replaced b y sections : let ( T , S ) be a section of G , minimal for the relation  , suc h tha t h T , S i G / ∈ I . Then by Prop osition 3.8, for a n y ( Y , X ) ∈ 40 Σ( G ) h T , S i G h Y , X i G = X g ∈ [ S \ G/X ] h T ∩ g Y , S ∩ g X i G ≡ X g ∈ G/X S ≤ g X T ≤ g Y h T , S i G (mo d . I ) = ψ T ,S  h Y , X i G  h T , S i G . Since I is prime, it follows that h Y , X i G − ψ T ,S ( h Y , X i G )1 Γ( G ) ∈ I . In partic- ular R = Γ( G ) /I is generated by the image o f 1 Γ( G ) , hence R ∼ = Z /p Z , where p is the characteristic of R . Since R is an in tegral domain, the n umber p is either 0 or a prime. 1. If p = 0, then R = Z , and ψ = ψ T ,S . And if ( T ′ , S ′ ) ∈ Σ( G ) is suc h that ψ T ,S = ψ T ′ ,S ′ , then b oth ψ T ,S  h T ′ , S ′ i G  and ψ T ′ ,S ′  h T , S i G  are no n zero. Then there exist elemen ts g , g ′ ∈ G suc h that ( T g , S g )  ( T ′ , S ′ ) and ( T ′ g ′ , S ′ g ′ )  ( T , S ), so ( T , S ) and ( T ′ , S ′ ) a re conjugate in G . 2. If p > 0, then R = Z /p Z , and ψ is equal to t he reduction of ψ T ,S mo dulo p . Since ψ  h T , S i G  = | N G ( T , S ) /S | is non zero in R , it follo ws that N G ( T , S ) /S is a p ′ -group. If ( T ′ , S ′ ) is another section o f G suc h that ψ is t he r eduction mo dulo p of ψ T ′ ,S ′ , and N G ( T ′ , S ′ ) /S ′ is a p ′ - group, then | N G ( T , S ) /S | = ψ T ,S  h T , S i G  ≡ ψ T ′ ,S ′  h T , S i G  (mo d . p ) . This is non zero. Similarly | N G ( T , S ) /S | ≡ ψ T ,S  h T ′ , S ′ i G  (mo d . p ) is non zero. In particular ψ T ′ ,S ′  h T , S i G  and ψ T ,S  h T ′ , S ′ i G  are b oth non zero, and it follows as ab o v e that ( T , S ) and ( T ′ , S ′ ) are conjugate in G . 14.3. Notation : L et p b e a prime numb er. • L et Σ p ( G ) denote the se t of se ctions ( T , S ) of G such that N G ( T , S ) /S is a p ′ -gr oup. In o ther wor ds Σ p ( G ) = Σ( G ) ∩ Π p ( G ) . • F or any ( T , S ) ∈ Σ( G ) , let ( T , S ) b b p denote the unique elemen t ( V , U ) of Σ p ( G ) , up to c on j uga tion , such that J T ,S,p = J V , U,p . 14.4. P r op osition : L et p b e a pri m e numb er. If ( T , S ) is a se ction of G , let ( T , S ) b + p denote a se ction of the form  P T , ( P S ) E P T  of G , wher e P is a 41 Sylow p -s ub g r o up of N G ( T , S ) . Define inductively an incr e asing se quenc e ( T n , S n ) in (Σ( G ) ,  ) by ( T 0 , S 0 ) = ( T , S ) , and ( T n +1 , S n +1 ) = ( T n , S n ) b + p , for n ∈ N . Then ( T , S ) b b p is c onjugate to the lar gest term ( T ∞ , S ∞ ) of the se quenc e ( T n , S n ) . Pro of : Again, t he pro of is the same as the pro of of Prop osition 7.4 : the section ( T , S ) b b p is a minimal elemen t ( V , U ) o f the p oset (Σ( G ) ,  ) suc h that ψ T ,S ( V , U ) = |{ g ∈ G/U | ( T g , S g )  ( V , U ) }| 6≡ 0 (mo d .p ) . Th us o ne can assume that ( T , S )  ( V , U ). But ψ T ,S ≡ ψ P T , ( P S ) E P T (mo d . p ) b y Corollary 4.4 and Lemma 11.2, for an y p -subgroup P of N G ( T , S ), hence one can a lso assume that ( T , S ) b + p  ( V , U ), hence that ( T ∞ , S ∞ )  ( V , U ), b y induction. Moreov er ψ T ∞ ,S ∞ ≡ ψ V , U (mo d . p ). As N G ( T ∞ , S ∞ ) /S ∞ is a p ′ -group, it follo ws that ( T ∞ , S ∞ ) = ( V , U ), as w as to b e shown. 14.5. P rop osition : L et ( T , S, p ) , ( T ′ , S ′ , p ′ ) b e elem ents of Ω( G ) . Then J T ′ ,S ′ ,p ′ ⊆ J T ,S,p if and onl y if • either p ′ = p and the se ctions ( T ′ , S ′ ) and ( T , S ) ar e c onjugate in G . • or p ′ = 0 and p > 0 , and the se ctions ( T ′ , S ′ ) b b p and ( T , S ) ar e c onjugate in G . Pro of : ( see Prop osition 7.6) Set J = J T ,S,p and J ′ = J T ′ ,S ′ ,p ′ . Then Γ( G ) /J ′ ∼ = Z /p ′ Z maps surjectiv ely to Γ( G ) /J ∼ = Z /p Z . Thus if p = p ′ , this pro jection map is an isomorphism, hence J = J ′ and the sections ( T , S ) and ( T ′ , S ′ ) are conj ug ate in G . And if p 6 = p ′ , then p ′ = 0 and p > 0. The morphism ψ T ,S is equal to the reduction mo dulo p o f the morphism ψ T ′ ,S ′ . In other w ords J T ′ ,S ′ ,p = J T ,S,p , hence ( T , S ) is conj ug ate to ( T ′ , S ′ ) b b p . 14.6. Corollary : L et p b e a prime numb er, and let Z ( p ) b e the lo c a li zation of Z at the set Z − p Z . L et Ω p ( G ) den ote the subset of Ω( G ) c o n sisting of triples ( T , S, 0) , for ( T , S ) ∈ Σ( G ) , and ( T , S, p ) , for ( T , S ) ∈ Σ p ( G ) . Then : 1. The pri m e ide als o f the ring Z ( p ) Γ( G ) ar e the ide als Z ( p ) J T ,S,q , for ( T , S, q ) ∈ Ω p ( G ) . 2. If ( T , S, q ) , ( T ′ , S ′ , q ′ ) ∈ Θ p ( G ) , then Z ( p ) J T ′ ,S ′ ,q ′ ⊆ Z ( p ) J T ,S,q if and only if : • either q = q ′ , and the se ctions ( T , S ) and ( T ′ , S ′ ) ar e c onjugate in G . • or q ′ = 0 , q = p , and the se ctions ( T ′ , S ′ ) b b p and ( T , S ) a r e c onjugate 42 in G . 3. The c onne cte d c omp onents of the sp e ctrum of Z ( p ) Γ( G ) ar e indexe d by the c onjugacy classes of Σ p ( G ) . The c omp onent indexe d b y ( T , S ) ∈ Σ p ( G ) c onsists o f a unique maxim al e lement Z ( p ) J T ,S,p , and of the ide als Z ( p ) J T ′ ,S ′ , 0 , wher e ( T ′ , S ′ ) ∈ Σ( G ) is such that ( T ′ , S ′ ) b b p is c onjugate to ( T , S ) in G . Pro of : (See Corollary 7.7) The prime ideals of Z ( p ) Γ( G ) are of the form Z ( p ) I , where I is a prime ideal of Γ( G ) suc h tha t I ∩ ( Z − p Z ) = ∅ . Equ iv alen tly I = J T ,S, 0 or I = J T ,S,p . This prov es Assertion 1. Now Assertion 2 follows from Prop osition 1 4.5, and Assertion 3 fo llows from Assertion 2. 14.7. Corollary : 1. The primitive idemp otents of the ring Z ( p ) Γ( G ) ar e indexe d by the c on- jugacy classes of Σ p ( G ) . The primitive idemp otent ε G V , U indexe d by ( V , U ) ∈ Σ p ( G ) is e qual to ε G V , U = X ( T ,S ) ∈ [Σ( G )] ( T ,S ) b b p = G ( T ,S ) γ G T ,S . 2. L et π b e a set of prime numb ers, and Z ( π ) b e the lo c alization of Z r elative to Z − ∪ p ∈ π p Z . L et F b e a s e t of se ctions of G , invariant by G - c onjugation, and [ F ] b e a set of r e pr esentatives of G - c onjugacy classes of F . Then the fol lowing c onditions ar e e q uivalent : (a) The idemp otent γ G F = X ( T ,S ) ∈ [ F ] γ G T ,S of Q Γ( G ) lies in Z ( π ) Γ( G ) . (b) L et ( T , S ) ∈ Σ( G ) , a nd let P b e a p -sub gr oup of N G ( T , S ) , for some p ∈ π . Then ( T , S ) ∈ F if and only i f  P T , ( P S ) E P T  ∈ F . Pro of : The pro o f is almost iden tical to the pro of of Corollary 7 .8 : let F b e a set of sections of G , in v ariant b y G -conjugation, and [ F ] b e a set of represen tativ es of G -conjugacy classes of F . The idemp otent γ G F = X ( T ,S ) ∈ [ F ] γ G T ,S 43 of Q Γ( G ) lies in Z ( p ) Γ( G ), for some prime p , if and only if there exists an in t eger m , not divisible b y p , suc h that u = mγ G F ∈ Γ( G ). Let ( T , S ) ∈ Σ( G ), and let P b e a p -subgroup of N G ( T , S ). The in teger ψ T ,S ( u ) is equal to m if ( T , S ) ∈ F , and to 0 otherwise. Hence it is coprime to p if a nd only if ( T , S ) ∈ F . Since ψ T ,S and ψ P T , ( P S ) E P T are congruen t mo dulo p , it follow s that ( T , S ) ∈ F if and only if  P T , ( P S ) E P T  ∈ F . Hence if ( T , S ) and ( T ′ , S ′ ) are sections o f G suc h that ( T , S ) b b p = G ( T ′ , S ′ ) b b p , then ( T , S ) ∈ F if and only if ( T ′ , S ′ ) ∈ F . Thus F is a disjoint union of sets of the form E V , U = { ( T , S ) ∈ Σ( G ) | ( T , S ) b b p = G ( V , U ) } , for some sections ( V , U ) ∈ Σ p ( G ). In other w ords the idempo t en t γ G F is a sum of some idemp oten ts ε G V , U , for ( V , U ) ∈ Σ p ( G ). But the primitiv e idemp oten ts of the ring Z ( p ) Γ( G ) are in one to one cor- resp ondence with the connected comp onen ts of its sp ectrum, whic h precisely are indexed b y the conjugacy classes of Σ p ( G ). It follow s that γ G F = ε G V , U is equal to the idemp oten t corresp onding to the comp onen t indexed b y ( V , U ), for an y ( V , U ) ∈ Σ p ( G ). This pro v es Assertion 1. This also prov es Assertion 2 in the case where π consists of a single prime n umber. F or t he general case, observ e that γ G F lies in Z ( π ) Γ( G ) if and only if it lies in Z ( p ) Γ( G ), for any p ∈ π . The following prop osition is the analogue of Theorem 7.9, for the ring Γ( G ). Its statemen t is a bit simpler : 14.8. Theorem : L et G b e a finite gr oup. 1. The primitive idemp otents of Γ( G ) ar e indexe d by the c onjugacy classes of p erfe c t sub gr oups of G . The idem p otent γ G H indexe d by the p erfe ct sub gr oup H is e q ual to γ G H = X ( T ,S ) ∈ [Σ( G )] D ∞ ( T )= G H γ G T ,S , wher e D ∞ ( T ) denotes the last term in the deriv e d series of T . 2. The prime sp e ctrum o f Γ( G ) is c onne cte d if an d onl y if G is solvabl e . Pro of : As in the pro o f of Theorem 7.9, let ∼ denote the finest equiv alence relation on the set Σ( G ) suc h that for any ( T , S ) , ( T ′ , S ′ ) ∈ Σ( G ) ∃ p, ( T , S ) b b p = G ( T ′ , S ′ ) b b p = ⇒ ( T , S ) ∼ ( T ′ , S ′ ) . 44 If w e can sho w that ( T , S ) ∼ ( T ′ , S ′ ) ⇔ D ∞ ( T ) = G D ∞ ( T ′ ) , then Assertion 1 follow s from Corolla ry 1 4.7, applied to the set π of a ll primes. Clearly , if there exists a p - subgroup P ≤ N G ( T , S ) such that ( T ′ , S ′ ) is conjugate to  P T , ( P S ) E P T  , then T ′ is conjugate to P T , and D ∞ ( T ′ ) is conjugat e to D ∞ ( T ). By transitivity , fo r any ( T , S ) , ( T ′ , S ′ ) ∈ Σ( G ), if ( T , S ) ∼ ( T ′ , S ′ ), t hen D ∞ ( T ) = G D ∞ ( T ′ ). T o sho w the conv erse, is it enough to show that ( T , S ) ∼  D ∞ ( T ) , D ∞ ( T )  , for an y ( T , S ) ∈ Σ( G ). Let p b e any prime, and P b e a Sylo w p -subgroup o f T . Then ( T , S ) b b p =  T , ( P S ) E P T  b b p , hence ( T , S ) ∼ ( T , S ′ ), where S ′ = ( P S ) E P T is a normal subgroup of T , con taining S , and of p ′ -index in T . Since p was arbitrary , it follows b y induction that ( T , S ) ∼ ( T , T ). No w for a n y prime p , if P is a Sylow p -subgroup of T , then P ≤ N G  O p ( T )  , a nd T = P O p ( T ). It follo ws that ( T , T ) b b p =  O p ( T ) , O p ( T )  b b p . Aga in, since p is arbitrary , it follows that ( T , T ) ∼  D ∞ ( T ) , D ∞ ( T )  , and this completes the pro of of Assertion 1 . Assertion 2 follow s easily , since b y Assertion 1, the sp ectrum of Γ( G ) is connected if and only if the trivial group is the only p erf ect subgroup of G , i.e. if G is solv able. 14.9. Corollary : T he ima ges of the primitive idem p otents of B ( G ) by the morphism i G : B ( G ) → Γ( G ) ar e the p rimitive idemp o tents of Γ( G ) . In other wor ds, if H is a p erfe ct sub gr oup of G , then γ G H = 1 | N G ( H ) | X K ≤ H | K | µ ( K , H ) h K , K i G . Pro of : Indeed b y a theorem of D ress ([6]), the primitiv e idemp oten ts of B ( G ) are indexed by the conjugacy classes of p erfec t subgroups of G . As the morphism i G : B ( G ) → Γ ( G ) is an injectiv e unital ring homomorphism (see Remark 10.7 and Prop osition 3 .1 1), it follow s tha t the primitive idemp o- ten ts o f B ( G ) are mapp ed to primitiv e idemp oten ts in Γ( G ), and t ha t ev ery primitiv e idemp oten t of Γ( G ) is in the image o f B ( G ). 15. Uni t g r o up All the results of Section 8 ab out the unit gr o up of the slice Burnside ring ha v e an analogue fo r the group Γ( G ) × of the section Burnside ring of a finite 45 group G . Namely : • The restricted ghost map yields an injectiv e group homomorphism Ψ × : Γ( G ) × ֒ → Y ( T ,S ) ∈ Σ( G ) Z × . The follow ing lemma follows : 15.1. Lemma : L et G b e a finite g r oup, and let u ∈ Γ( G ) . The fol lowing c onditions ar e e quival e nt : 1. u ∈ Γ( G ) × . 2. ψ T ,S ( u ) ∈ {± 1 } , for any ( T , S ) ∈ Σ( G ) . 3. u 2 = 1 . In p articular Γ( G ) × is a finite elemen tary ab elian 2-gr oup. • The follo wing is an analog ue of Prop osition 8 .3 : 15.2. Prop osition : F eit-Thompson ’s the or em is e quivalen t to the state- ment that, if G has o dd or der, then Γ( G ) × = {± 1 } . Pro of : The pro o f is the same as the pro of of Prop osition 8.3 : the argu- men t uses only the for m ulae for the primitiv e idemp oten ts o f Q Γ( G ) (Corol- lary 12.5), and the c ha racterization of solv able groups b y the connectednes s of the prime sp ectrum of Γ( G ) (Prop osition 14.8). • The follo wing theorem is an analogue of Y oshida’s c haracterization ([10]) of the unit group of the usual Burnside ring : 15.3. Theorem : L et G b e a finite gr o up, and let m =  m ( T , S )  ( T ,S ) ∈ Σ( G ) b e a se quenc e of in te ge rs in {± 1 } indexe d by Σ( G ) , c onstant on G -c onjugacy classes of se c tion s . Then the se quenc e [ m ] =  m ( T , S )  ( T ,S ) ∈ [Σ( G )] of r epr e- sentatives li e s in the im age of the r e stricte d ghost map Ψ × if and only if for any ( T , S ) ∈ Σ( G ) , the map g ∈ N G ( T , S ) /S 7→ m  , E  /m ( T , S ) ∈ {± 1 } is a gr oup homom orphism. 46 Pro of : Again, the pro o f is the same as for Theorem 8.4 : it only requires Theorem 13.1 and Lemma 11.2. • Finally , the corresp ondence G 7→ Γ( G ) × is not a biset functor : 15.4. Prop osition : The c orr es p ondenc e sending a finite gr oup G to Γ( G ) × c annot b e endowe d with a structur e of biset functor. Pro of : The pro of of Prop osition 8.6 g o es t hrough without c hang e here : the arg umen t uses computations for the trivial group, the group C 2 , and the group ( C 2 ) 2 . All these groups are ab elian, hence sections a nd slice s coincide. App endix 16.1. Let G and H be finite groups, and U b e a finite ( H , G ) - biset. Let U op denote the opp osite biset , i.e. the ( G, H )-biset equal to U as a set, with actions rev ersed by taking in v erses (i.e. g · u · h [in U op ] = h − 1 ug − 1 [in U ], for g ∈ G , u ∈ U , and h ∈ H ). When X is a finite G - set, the set Hom G - set ( U op , X ) is a finite H -set : if ϕ : U op → X is a morphism of G -sets, and if h ∈ H , then the morphism hϕ : U op → X is defined b y ( hϕ )( u ) = ϕ ( h − 1 u ), for u ∈ U . This corresp ondence T U : X 7→ Hom G - set ( U op , X ) is actually a functor from the category G - set of finite G - sets to the category H - set . One can sho w (see e.g. Section 11.2 .13 o f [5]) that this functor induces a map t U : B ( G ) → B ( H ) b et w een the usual Burnside rings of G and H , called the gener alize d tensor induction with resp ect to U . This induction is no t additiv e, but m ultiplicativ e (i.e. t U ( ab ) = t U ( a ) t U ( b ), fo r an y a, b ∈ B ( G )). It yields a biset functor structure on the unit group of the usual Burnside ring. A natural question is to kno w whether this construction can b e extended to the rings Ξ( G ) and Γ( G ) : indeed, if X f → Y is a morphism of finite G -sets, then the morphism T U ( X f → Y ) Hom G - set ( U op , X ) Hom G - set ( U op ,f ) / / Hom G - set ( U op , Y ) is a morphism of finite H -sets. Do es this correspo ndence induce a map Ξ( G ) → Ξ( H ) or a map Γ( G ) → Γ( H ) ? In other words : 47 (Q1) Are the defining relations of Ξ( G ) preserv ed b y T U ? (Q2) Do es T U map a Galois morphism to a Galo is morphism ? The answ er t o these tw o questions has to b e no in general, for otherwise T U w ould yield a biset functor structure on the unit groups o f the slice Burnside ring and the section Burnside ring, contradicting Prop osition 8.6 and Prop o- sition 15.4. But the answ er is y es with the additional assumption that the biset U b e left inert , according to the following definition : 16.2. Definition : A n ( H , G ) -bi s et U is c al le d left inert if H u ⊆ u G , fo r any u ∈ U , in other wor ds if H a c ts trivial ly on the set of orbits U / G . 16.3. E xample : • If U is transitiv e a s a rig h t G -set, then U is left inert, since | U /G | = 1. In particular, the iden tit y ( G, G )-biset G is left inert. Conv ersely , if U is left inert, t hen eac h righ t orbit uG , for u ∈ U , is an ( H , G )-biset, so U is a disjoin t union of ( H , G )- bisets whic h are transitiv e as righ t G -sets. • The disjoin t union of tw o left inert ( H, G )-bisets is left inert. An y sub-biset of a left inert biset if left inert. • Left inert bisets can b e comp osed : if G , H , a nd K are groups, if U is a left inert ( H , G )-biset, and if V is a left inert ( K , H )-biset, then V × H U is a left inert ( K, G )-biset : indeed, fo r u ∈ U and v ∈ V K ( v , H u ) = ( K v , H u ) ⊆ ( v H , H u ) = ( v , H H u ) ⊆ ( v , H uG ) = ( v , H u ) G , where ( v , H u ) denotes the image of ( v , u ) in V × H U . 16.4. The follo wing prop osition deals with Question (Q2) ab ov e, in the case of a left inert biset : 16.5. Proposit ion : L et X f → Y b e a Galois mo rp hism of G -s e ts. I f U is a left inert ( H , G ) -biset, then the morphi s m T U ( X f → Y ) is a Galois m orphism of H -se ts. Pro of : Let ϕ, ψ ∈ T U ( X ) = Ho m G - set ( U op , X ) ha ving the same image in T U ( Y ). This means that f ◦ ϕ = f ◦ ψ , i.e. that f  ϕ ( u )  = f  ψ ( u )  , for any u ∈ U . Since f is a Galois morphism, it f ollo ws tha t G ϕ ( u ) = G ψ ( u ) , for an y u ∈ U . 48 Supp ose that h ∈ H stabilizes ϕ . It means that ϕ ( h − 1 u ) = ϕ ( u ), for any u ∈ U . But since H u ⊆ u G , there exists g ∈ G (dep ending on u ) suc h that hu = ug , i.e. h − 1 u = ug − 1 . Then ϕ ( u ) = ϕ ( h − 1 u ) = ϕ ( ug − 1 ) = g ϕ ( u ) , hence g ∈ G ϕ ( u ) . It follo ws that ψ ( u ) = g ψ ( u ) = ψ ( ug − 1 ) = ψ ( h − 1 u ) = ( hψ )( u ) . Hence h stabilizes ψ , and by symmetry ϕ and ψ ha v e the same stabilizer in H . This sho ws that T U ( X f → Y ) is a Galo is morphism of H -sets. 16.6. R emark : The f ollo wing example sho ws that Lemma 16.5 is no longer true without the hypothesis that U is left inert : supp ose that G = 1 , and that H is non trivial. Let U = H , view ed as an ( H , G ) - biset. Then U is not left inert. Let X b e a set of cardinality 2, let Y b e a set of cardinality 1, and let f : X → Y b e the unique map. Then f is a Galo is morphism of G - sets, for trivial reasons. In this case Hom G ( U op , X ) is isomorphic to the set 2 H of subsets of H , on whic h H acts by left translation. The set Hom G ( U op , Y ) is a set • of cardinalit y 1, and the map T U ( f ) is the o nly p ossible map 2 H → • . In particular, all the elemen ts of 2 H ha v e the same image. But the subset { 1 } of H has a trivial stabilizer, whereas the subset H of H has stabilizer equal to H . Th us T U ( X f → Y ) is not a Galois morphism of H - sets, b y Prop osition 9.3. 16.7. R emark : The example giv en in R emark 16.6 also sho ws that the answ er to Question (Q1 ) is no, in g eneral : indeed, k eeping the same notation, the image of the morphism X f → Y in Ξ( G ) is equal to the sum of tw o copies of the image of Y Id → Y . In o ther w o rds, in Ξ( G ) ∼ = Z , π ( X f → Y ) = 2 = π ( X Id → X ) . But T U ( X f → Y ) is isomorphic to 2 H → • , hence ( 16 . 8 ) π  T U ( X f → Y )  = X A ⊆ H A mod . H h H , H A i H b y Lemma 3.4, where the summation runs ov er a set of represen tatives o f subsets of H , up t o tra nslation b y H , and H A denotes the stabilizer in H of 49 suc h a subset A . In particular H A = H if and only if A = H o r A = ∅ , hence if K is a subgroup of H , the only term of the f orm h K, K i H in the righ t hand side of Equation 16.8 is h H , H i H , with co effic ien t 2. On the o ther hand, b y Lemma 3.4 again π ( X Id → X ) = X A ⊆ H A mod . H h H A , H A i H . As there are prop er non empt y subsets A of H , there are some terms of the form h K, K i H with non zero co efficien t in this summation, for some subgroups K < H . Hence π ( X f → Y ) 6 = π ( X Id → X ), thus T U do es not preserv e the defining relations of Ξ( G ). 16.9. The follo wing prop osition answe rs Question (Q1) in the case of left inert bisets : 16.10. Prop osition : 1. L et G and H b e fin i te gr oups, and U b e a finite left inert ( H , G ) -bis et. Then T U induc es a wel l defi ne d map t U : Ξ( G ) → Ξ( H ) , such that t U  Γ( G )  ⊆ Γ( H ) . Mor e over t U only dep ends on the isomorphi sm class of the biset U . 2. The map t U is multiplic a tive, i.e. t U ( ab ) = t U ( a ) t U ( b ) , for any a, b ∈ Ξ( G ) . Mor e over t U (1 Ξ( G ) ) = 1 Ξ( H ) . In p articular t U r estricts to gr oup homomorphism s Ξ( G ) × → Ξ( H ) × and Γ( G ) × → Γ( H ) × . 3. If U a n d U ′ ar e fin i te left inert ( H , G ) - b isets, then t U ⊔ U ′ = t U t U ′ . 4. If G , H , and K ar e finite gr oups, if U is a finite left inert ( H , G ) -biset and V is a finite le f t inert ( K , H ) -biset, then t V ◦ t U = t V × H U . 5. If U is the identity ( G, G ) -biset, then t U is the i d entity ma p. Pro of : Since U is left inert, Example 16.3 sho ws t ha t U splits as a disjoin t union of ( H , G )- bisets U ∼ = G u ∈ [ U /G ] uG . It follows that the functor T U = Hom G - set ( U, − ) is isomorphic to the direct pro duct of the functors T uG , for u ∈ [ U / G ]. Hence to prov e Assertion 1, it suffices to consider the case where U is transitiv e as a right G - set. In this cas e, the functor T U induces actually a gr oup homom orphism f r o m Ξ( G ) to Ξ( H ) : to see this, it suffices to che c k that the defining relatio ns of Ξ( G ) are mapp ed to relatio ns in Ξ( H ). Fix u ∈ U , and denote b y G u its 50 stabilizer in G . So U = u G ∼ = G/G u as G -set, and f o r a n y G -set X , there is a bijection Hom G - set ( U op , X ) ∼ = X G u . This is actually an isomorphism of H -sets if the left a ction of H on X G u is defined as fo llows : f or h ∈ H , there is some g ∈ G suc h that hu = u g , and the action of h on X G u is defined b y hx = g − 1 x , for x ∈ X G u . No w let X f → Y b e a morphism of finite G -sets such that X splits as a disjoin t union X = X 1 ⊔ X 2 of tw o G - sets. The image π ( X f → Y ) of this morphism in Ξ( G ) is equal to the sum o f the images of the morphisms X 1 f 1 → f ( X 1 ) and X 2 f 2 → f ( X 2 ) , where f 1 and f 2 are the restrictions of f to X 1 and X 2 , resp ectiv ely . On the other hand T U ( X f → Y ) is isomorphic t o X G u 1 ⊔ X G u 2 f G u − → Y G u in the catego ry H - Mo r , where f G u is the restriction of f to X G u . The image π  T U ( X f → Y )  of this morphism in Ξ( H ) is equal to the sum of t he images of the morphisms X G u 1 f G u 1 − → f ( X G u 1 ) and X G u 2 f G u 2 − → f ( X G u 2 ) . By Lemma 3.3, since f ( X G u 1 ) ⊆ f ( X 1 ) G u π  X G u 1 f G u 1 − → f ( X G u 1 )  = π  X G u 1 f G u 1 − → f ( X 1 ) G u  , whic h is equal to π  T U  X 1 f 1 → f ( X 1 )   . It follo ws that in Ξ( H ) π  T U ( X f → Y )  = π  T U  X 1 f 1 → f ( X 1 )   + π  T U  X 2 f 2 → f ( X 2 )   , hence T U induces a group homomorphism t U : Ξ( G ) → Ξ( H ). As the func- tor T U maps direct pro duc t o f G -sets to direct pro duct of H - sets, and the trivial G -set to the trivial H - set, t he mo r phism t U is actually a unital rin g homomorphism from Ξ( G ) to Ξ( H ) (recall that U is assumed transitive as a righ t G -set, here). If U is an arbitrary finite left inert ( H , G )-biset, then T U induces the map t U = Y u ∈ [ U /G ] t uG : Ξ( G ) → Ξ( H ) . It follow s from Prop o sition 16.5 that t U  Γ( G )  ⊆ Γ( H ). 51 No w Assertion 2 fo llo ws, since t U is equal to a pro duct of unital ring homomorphisms. Assertions 3 , 4 , and 5 are straightforw ard consequences of the prop erties of the functor T U . 16.11. Remark : It follo ws that the corresp ondences G 7→ Ξ( G ) × and G 7→ Γ( G ) × are biset functors for whic h the biset op erations are only defined for left inert bisets . Equiv alen tly , these corresp ondences a r e biset functors without induction : the usual basic op erations for biset functors are defined for these corresp ondences, namely r e s triction to a subgroup, deflation from G t o a factor group G/ N (this is induced b y ta king fixe d p oints by N on G -sets), tr ansp ort by isomorphism, and inflatio n from a factor group. But there is no induction from a subgroup. 16.12. The last result of this App endix is the computation of the gro up Ξ( G ) × , when G is ab elian : 16.13. Theorem : L et G b e a finite ab elian gr oup. T hen Ξ ( G ) × has an F 2 -b asis c onsisting o f the elemen ts   −h G, G i G h G, G i G − h S, S i G h G, G i G − h G, S i G  for | G : S | = 2 . In p articular Ξ( G ) × ∼ = ( F 2 ) 2 r +1 , wher e r is the numb er of sub gr o ups of index 2 in G . Pro of : By Theorem 8.4, the group Ξ( G ) × is isomorphic to the group of sequence s ( u T ,S ) ( T ,S ) ∈ Π( G ) with v alues in {± 1 } , suc h that for any ( T , S ) ∈ Π( G ), the map g ∈ G/S 7→ u , /u T ,S is a group homomorphism. Switc hing to an additiv e notation, the g roup Ξ( G ) × is isomorphic to the F 2 -v ector spaces of sequences ( λ T ,S ) ( T ,S ) ∈ Π( G ) , with v alues in F 2 , suc h that for an y ( T , S ) ∈ Π( G ) and any ( g , h ) ∈ G λ , + λ , + λ , + λ T ,S = 0 . If none of g , h, and g h are in S , this yields an expression λ T ,S = λ , + λ , + λ , of λ T ,S as a linear com bination of λ T ′ ,S ′ , for S ′ > S . If | G : S | > 2, it is alw ays p ossible to find suc h elemen t s g a nd h . It follow s that the sequence 52 ( λ T ,S ) ( T ,S ) ∈ Π( G ) is entire ly determined b y the v alues λ T ,S , fo r | G : S | ≤ 2, i.e. the v alues λ G,G , λ S,S and λ G,S , where S runs through the set of subgroups of index 2 in G . If there are r suc h subgroups, this g iv es 2 r + 1 suc h v alues, so dim F 2 Ξ( G ) × ≤ 2 r + 1. T o prov e the con v erse, and actually the more precise statement in the theorem, it suffices to show that the eleme n ts − 1 Ξ( G ) , u G S = h G, G i G − h S, S i G and v G S = h G, G i G −h G, S i G , for | G : S | = 2 are linearly indep endent elemen ts of the F 2 -v ector space Ξ( G ) × . First a s h G, G i G = 1 Ξ( G ) , and as h S, S i 2 G = 2 h S, S i if | G : S | = 2, by Prop osition 3.8, it follows that ( u G S ) 2 = ( v G S ) 2 = 1, hence { u G S , v G S } ⊆ Ξ( G ) × . Observ e no w that for | G : S | = 2, b oth elemen ts u G S and v G S are o btained b y inflation from G/S to G o f the cor r espo nding elemen ts u G/S 1 and v G/S 1 . The group C = G/S has order 2. The additiv e group Ξ( C ) has a Z -basis consisting of h C , C i C , a = h C, 1 i C and b = h 1 , 1 i C . The elemen t h C , C i C is the iden tity elemen t 1 Ξ( C ) of Ξ( C ), and by Prop osition 3.8, the pro ducts of the ot her ba sis elemen ts are giv en b y a 2 = 2 a , b 2 = ab = ba = 2 b . The group of units Ξ( C ) × has an F 2 -basis consisting of − 1 Ξ( C ) , x = 1 Ξ( C ) − a and y = 1 Ξ( C ) − b (a ctually Ξ( C ) is e qual to the gr o up algebra of the m ultiplicativ e group < − 1 Ξ( C ) , x, y > ∼ = ( C 2 ) 3 ). Moreov er, the set { x, y } is an F 2 -basis of the k ernel ∂ Ξ( G ) × of the deflation map Def C C /C : Ξ( C ) × → Ξ( 1 ) × = {± 1 Ξ( 1 ) } (recall that deflation in this case consists in taking fixed p oin ts under C ). No w assume tha t in Ξ( G ) × , there is a linear relation (with an additiv e notation) of the form λ ( − 1 Ξ( G ) ) + X | G : S | =2 ( α S u G S + β S v G S ) = 0 , for some co effi cien ts λ, α S , β S in F 2 . F ix a subgroup X of index 2 in G , and apply Def G G/X to this relation. Observ e that Def G G/X u G S = Def G G/X Inf G G/S u G/S 1 = Inf G/X G/S X Def G/S G/S X u G/S 1 is equal to 0 if S 6 = X , and to u G/S 1 if S = X . Similarly D ef G G/X v G S is equal to 0 if S 6 = X , a nd to v G/S 1 if S = X . It fo llo ws that λ ( − 1 Ξ( G/X ) ) + α X u G/X 1 + β X v G/X 1 = 0 , hence λ = α X = β X = 0 . This completes the pro o f , since this holds for an y X of index 2 in G . 16.14. Remark : Using the inclusion i G : B ( G ) → Ξ ( G ) of Prop osi- tion 3.1 1, one can iden tify B ( G ) with a subring of Ξ( G ). Then it is easy to 53 see that B ( G ) × is the subgroup of Ξ( G ) × generated b y the elemen t s −h G, G i G and h G, G i G − h S, S i G , for | G : S | = 2 : this giv es another pro of of Matsuda’s theorem ([8]), saying t ha t the unit group B ( G ) × of the usual Burnside ring of a finite a b elian g r oup G is isomorphic to ( F 2 ) r +1 , where r is the nu m b er of subgroups of index 2 in G . References [1] R. Boltje and G. Pfeiffer. An algorithm for the unit group of the Burnside ring o f a finite g roup. In Gr oups S t Andr ew s 2005 , volume 33 9, pages 230–236. London Math. So c. Lectures Notes Series, 20 07. [2] S. Bo uc. Construction de foncteurs en tre cat ´ egories de G -ensem bles. J. of Algebr a , 1 83(0239):7 37–825, 19 9 6. [3] S. Bouc. Burnside rings. In Han d b o ok of Algebr a , v olume 2, chapter 6E, pages 739–804. Elsevier, 2000. [4] S. Bouc. The functor of units of Burnside rings for p -gro ups. Comm. Math. Helv. , 82:5 83–615, 2007. [5] S. Bouc. B i s e t functors for finite gr oups , volume 1990 o f L e ctur e Notes in Mathematics . Springer, 2010. [6] A. Dress. A c haracterization of solv able groups. Math. Zeit. , 110:2 1 3– 217, 1969. [7] The GAP Group. GAP – Gr oups, A lgo- rithms, and Pr o gr amming, V ersion 4.4.1 2 , 2008. (\protect\v rule width0pt\protect \href{http://www.gap-system.org}{http://www.g a p - s y s t e m . o r g } ) . [8] T. Matsuda. On the unit g r o up of Burnside rings. Jap an. J. Math. , 8(1):71–93 , 1982. [9] T. tom Diec k. T r ansformation gr oups an d r epr esen tation the ory , v olume 766 of L e ctur e Notes in Mathematics . Springer- V erla g , 1979. [10] T. Y oshida. On the unit groups of Burnside rings. J. Math. So c. Jap an , 42(1):31–6 4, 1990 . Serge Bouc - CNRS-LAMF A, Univ ersit ´ e de Picardie, 33 rue St Leu, 80 0 39, Amiens Cedex 01 - F rance. email : serge.bouc@u-picar die.fr http://www. lamfa.u-picardie.fr/bouc/ 54