Bounds for Bilinear Complexity of Noncommutative Group Algebras

Bounds for B ilinear Complexit y of Noncomm utativ e Grou p Algebras Alexey Pospelov Computer Science F acult y , Saarland Universit y , pospelov@c s.uni-saarla nd.de Abstract. W e study the complexity of multiplication in n oncomm uta- tive group algebras which is closely related to the complexit y of matrix multipli cation. W e characteri ze such semisimple group algebras of t he minimal bilinear complexity and show non trivial low er b ounds for the rest of th e group algebras . These lo wer bound s are built on the top of Bl¨ aser’s results for semisimple algebras and algebras with large radical and the low er b ound for arbitrary associative algebras due to A lder and Strassen. W e also sho w sub quadratic upp er bou n ds for a ll group algebras turning into “almost linear” provided the exp onent of matrix multipli- cation eq uals 2. 1 In tro duction W e study noncomm utative group algebras and the problem of computing the prod uct of tw o elements of an algebra. W e restrict ourselves on t he so-called rank or bilinear complexity of multiplication, which, roughly sp eak in g, counts only the bilinear multiplicatio ns u sed by an algo rithm, i.e. multiplications where eac h of the op erands depen ds on one of th e input vectors. A quadratic (in terms of dimension of an algebra) up p er b ound is straightforw ard, while all curren tly know n general low er b oun ds are linear. This researc h is motiv ated by the recent group-th eoretic approach for ma- trix multiplicatio n b y Cohn a nd U mans [9 ] and follo wing group- theoretic algorithms for matrix multiplication [10]. It was sho wn th at finite groups p ossessing some sp ecial properties can b e used to desi gn effectiv e matrix multipli cation algorithms. Ou r goal is to explore the structu re of group algebras and inv estigate structural and complexity relation b etw een non- comm utative group algebras and the matrix algebra. W e inves tigate this approac h and put it into a d ifferent ligh t. In fact, w e show that the group algebras for th e most promis ing groups for the group-theoretic app roac h hav e essen tially th e same complexity as the matrix multiplication itself. On the other hand, for a wide class of group algebras a low er b ound holds which dep ends on the exp onent of matrix multiplica tion ( denoted in literature by ω , see Sect. 3 for defin ition). I f one finds a more effec- tive algorithm of multiplicatio n in these group algebras, it would giv e a b etter upp er b ound for ω (but without necessary proving ω = 2, which is th e general conjecture [6]). W e also study general bilinear complexit y of noncommutativ e group algebras and this pap er extend s the researc h in [22,23,7] where the problem for comm utative group algebras ov er arbi- trary fields w as solv ed entire ly . Our results also improv e t he Atkinson’s upp er b ound for the total complexit y of multiplication in group alge- bras [2]. Using Bl¨ aser’s th eorem on classification of all algebras of the minimal rank (see Sect. 5 ) w e form ulate a criterion for a semisimple group al- gebra to b e an algebra of th e minimal b ilinear complexit y . F or some sp ecial cases w e also show a 5 2 · dimension-lo w er b ounds for the rank of group algebras. F or other sp ecial cases we show an u p to 3 · dimension of an algebra lo wer b ound . F or one special class of groups having not “too man y” different irreducible representatio ns we sho w a low er b ound whic h dep end s on the exp onent of matrix multiplications and turns to b e sup erlinear if the exp on ent of matrix multiplic ation do es not equal to 2. This employs Sch¨ onhage’s τ -theorem ( see Sect. 5). W e sho w that this class is not empty , for in stance group algebras of symmetric groups of order n ! and general linear groups o ver finite fields hav e such a low er b ound. Another motiv ation for this work was the searc h for algebras of high bilinear complexity . It is known, th at ov er algebraically closed fields there exist families of algebras of arbitrarily high dimensions with bi- linear complexity of each algebra from t he family strictly greater t h an (dimension of the algebra) 2 27 [6, Exercise 17.20]. How ever, no concrete exam- ples are kn own. This is in some sense similar to the situation in logical synthesis theory , where it is know n that the circuit complexity (in a full basis) of almost all b o olean functions of n vari ables is asymptotically c 2 n n [21] where the constant c dep ends solely on the b asis, e.g. for the classical circuit b asis {∨ , & , ¬} , c = 1. 1 But there is no explicit con- struction of a function of n v ariables with a sup erlinear low er b ound on the num b er of gates in a full fin ite functional basis. W e show th at a broad class of group algebras h as sup erlinear bilinear complexity if th e exp onent of matrix multiplication do es n ot equal to 2. W e then turn to upper bound s and show by a simple tec h nique a general upp er bound for the total complexity of m u ltiplication in group al gebras that depend s on th e total complexity of matrix m ultiplication. I n fact, if the exp onent of matrix multiplication equals 2, then the total complexity of the multiplication in group algebras is alw a ys “almost linear”. W e indicate some special cases, when this up p er b ound can b e impro ved provided a maximal irreducible represen tation of the group has not t oo high dimension. F or lo we r b ounds w e distinguish b etw een the semi simple and th e mo d- ular case. If the characteristic of the ground field is either zero or d oes not divide t he order of the group then the group algebra is kn o wn to b e semisimple. In the other case, if the characteristic p d ivides the order of the group, then the algebra h as nontrivial radical. In some cases its structure inside the group algebra can be d escribed exactly . But in gen- eral this introdu ces additional significan t difficulties. If the radical has relativ ely small nilp otence index t hen it is p ossible to obtain relativ ely 1 In fact, for a full circuit basis B = { f 1 , . . . , f n } where each f ν is of m ν v ariables (with no fictitious dep endenies) and h as wei ght w ν , the constant c = min 1 6 ν 6 n m ν > 2 w ν m ν − 1 . 2 high lo wer boun ds for the bilinear complexity of m ultiplication in group algebra. Finally , w e show direct relations b etw een complexit y of noncommutativ e group al gebras and co mplexity of matrix multiplicatio n and pose sev eral open questions. The paper is organized as follo ws: in Sect. 2 w e bring all necessary def- initions an d notions from algebra and representation theory . In S ect. 3 w e in t rodu ce the mod el of computation w e will b e w orking with and for- mula te related computational problems. W e discuss briefly tight relation b etw een differen t algebraic notions and compu tational complexity . W e introduce an imp ortant quan titative measure estimate for complexity of multipli cation in families of algebras of gro wing dimensions which gen- eralizes the well-kno wn notion of the exp onent of matrix multiplicati on. Classica l structural results from the theory of finite-dimensional algebras and representation theory will b e presented in Sect. 4. Section 5 con tains all necessary results from the algebraic complexity theory to b e emplo yed for obtaining lo we r and up p er b ounds for the complexity of multipli cation in group algebras. In S ect. 6 we prov e the first part of our main result. W e sho w, that for any “complicated enough” group its corresponding group algebra is not of the minimal rank. W e also pro ve tw o differen t kinds of lo w er bounds f or fa milies of group algebras depending on the representa- tions of their groups. W e also show the general relation b etw een the low er b ound for the complexity of group algebra multiplication and the com- plexity of matrix multiplic ation. W e show , th at the bilinear complexity of m u ltiplication in group algebras of sym met ric groups is superlinear in their dimension if th e exp on ent of matrix multiplic ation do es not equal 2. In Sect. 7 we turn to effective algorithms for multiplication in group algebras. W e sh ow the general upp er b ound for multiplic ation in any group algebra d ep en ding on the ex p onent of matrix multiplica tion and some improve ments based on particular prop erties of the group. 2 Basic Definitions In what follo ws w e alw ays use the term algebr a for an associative al- gebra with u nity . F or example, n × n -matrices ov er some field form an algebra, and so do univ ariate p olynomials ov er some fi eld modulo some fixed p olynomial or multiv ariate p olynomials mo dulo some system of p olynomials. A b asis of an algebra is any b asis of the u nderlying vector space. The dimension (dim A ) of an algebra A is the d imension of the underlyin g vector space. The m ultiplication in an algebra is completely defined if it is d efined for th e vectors of any of its bases: let A b e an algebra ov er k , n = d im A , and e 1 , . . . , e n b e some b asis of A , then e i · e j = n X ν =1 α ν ij e ν , 1 6 i, j 6 n, where α ν ij are the structur al c onstants from th e field k . W e call a b asis { e i } n i =1 of A a gr oup b asis if the vectors e i form a multiplicativ e group 3 with respect to the multiplication in algebra. In this case A is called a gr oup algebr a . On the other hand, given a fi nite gro up G = { g 1 , . . . , g n } and a field k we can define a group algebra k [ G ] as a n -dimensional vector space over k with basis { g i } n i =1 and multiplicatio n in k [ G ] defined as n X i =1 α i g i ! · n X j =1 β j g j ! = n X ℓ =1 g i g j = g ℓ α i β j g ℓ . W e call the dir e ct pr o duct of th e algebras A and B ov er one and the same field k the algebra A × B ov er k which consists of p airs of v ec- tors ( a, b ) , a ∈ A, b ∈ B and all op erations in A × B are performed compon ent-wise: ( a 1 , b 1 ) ◦ ( a 2 , b 2 ) = ( a 1 ◦ a 2 , b 1 ◦ b 2 ), ◦ ∈ { + , − , ·} and λ · ( a, b ) = ( λ a, λb ), where a i ∈ A, b i ∈ B , i = 1 , 2 , λ ∈ k . W e call B ⊆ A a sub algebr a of A , if B is a linear subspace of A an d the prod uct (in A ) of any tw o vectors of B lies in B . A subalgebra I of A is called l eft ( ri ght ) i de al of A if for all a ∈ A , x ∈ I the pro duct ax ∈ I ( xa ∈ I resp.) A left ideal that is at the same time a rig ht ideal is called a two-side d ide al . A ( left, righ t, tw o-sided) ideal is called m aximal if it is not contained in any other prop er (left, right, tw o-sided) ideal of the algebra. An ideal I is called nilp otent if I m = { 0 } for some m > 0. 2 The smallest m with t his p rop erty is called th e ni lp otenc e i ndex of I . The sum of all nilp otent left ideals of an algebra A is called the r adic al of A and is denoted by rad A . The intersection of all the maximal left ideals of the algebra A is called the Jac obson r adic al of A and is denoted by J ( A ). Proposition 1. L et A b e an algebr a over field k . Then rad A = J ( A ) . Pr o of. This follo ws from the fact, t h at the desc ending chain c ondition for left ideals in A implies rad A = J ( A ), see [26]. It ensures that any fa mily of left ideals in A contains at least one minimal ideal, i.e. an ideal t h at does not contain any other id eal of th e family . In a finite-d imensional algebra this alw a ys holds since we can map any family of ideals to th e subset of integers in [0 , dim A ] mapping each ideal to its dimension as a linear subspace. The resulting image will contai n the minimal element whic h will correspond to th e set of id eals from the family having the minimal dimension. Obviously , any of these is minimal. ⊓ ⊔ The nilp otence index of rad A will b e denoted by N ( A ). The set of all x ∈ rad A such that x · rad A = { 0 } is called the left annihilator of rad A and is d enoted by L A . The right annihi lator R A is introdu ced in the similar mann er. Algebra A is called a division algebr a if every element of A h as an in- verse in A with resp ect to the multiplication in A . A is called lo c al if A/ rad A is a division algebra, and A is called b asic if A / rad A is a direct prod uct of division algebras. F ollowing Bl¨ aser [5] w e call A sup erbasic if A/ rad A ∼ = k t for some t > 1. 2 F or a set S with multiplication and a positive intege r r S r denotes the set of all p ossible pro ducts of r elements of S : { s 1 · · · s r : s ρ ∈ S, 1 6 ρ 6 r } . 4 Algebra A is called se misimpl e if rad A = 0 and simple if it do es not con- tain any prop er tw osided idea ls except for the { 0 } . Structure of semisim- ple and simple algebras is describ ed in W edderburn’s theorem whic h can b e found in [26]. Theorem 1. Every finite dimensional semisimple al gebr a over some field k is i somorphic to a finite dir e ct pr o duct of si mple algebr as. Every finite dimensional simple k -algebr a A is isomorphic to an algebr a D n × n for an inte ger n > 1 and a k -division algebr a D . The inte ger n and the algebr a D ar e uni quely determine d by A ( the latter up to isomorphism ) . 3 Computational Mo del Let U, V , and W b e fin ite d imensional vec tor spaces o ver a field k . Let ϕ : U × V → W b e a b ilinear map. A bil ine ar algorithm for ϕ is a se- quence ( u 1 , v 1 , w 1 ; . . . ; u r , v r , w r ) where u ρ ∈ U ∗ , v ρ ∈ V ∗ , w ρ ∈ W such that for all x ∈ U, y ∈ V ϕ ( x, y ) = r X ρ =1 u ρ ( x ) v ρ ( y ) w ρ . r is called the length of the bilinear algorithm and the minimal length ov er all bilinear algorithms for ϕ is called the r ank or the bili ne ar c omplexity of ϕ and is den oted by rk ϕ . A sequence ( u 1 , v 1 , w 1 , . . . , u ℓ , v ℓ , w ℓ ) where u λ , v λ ∈ ( U × V ) ∗ , w λ ∈ W such th at for all x ∈ U, y ∈ V ϕ ( x, y ) = ℓ X λ =1 u λ ( x, y ) v λ ( x, y ) w λ is called a quadr atic alg orithm for ϕ . ℓ is called t he length of the q uadratic algorithm an d the minimal length o ver all quadratic algorithms for ϕ is calle d th e mul tiplic ative c om pl exity of ϕ and is d enoted by C ( ϕ ). Obviously C ( ϕ ) 6 rk ϕ . A straightforw ard argument implies also that rk ϕ 6 2 C ( ϕ ) and except for t rivial cases, rk ϕ < 2 C ( ϕ ) [15]. Multiplication in algebra A is a bilinear map. Rank and multiplicativ e complexity of m ultiplication in A are called r ank and multipli c ative c om- plexity of A and are d en oted by rk A and C ( A ) respectively . Obviously , rk A × B 6 rk A + rk B (also C ( A × B ) 6 C ( A ) + C ( B )). How ever, it is not known if t he conve rse also holds whic h is k now n as the famous Strassen’s Direct Sum Conjecture [6, p. 360]. Obviously , rank (and therefore, m ultiplicative complexity) of an y algebra A is at most (d im A ) 2 . Let A = { A 1 , A 2 , . . . } b e a family of algebras over a field k . W e define ω A , the r ank-exp onent of multipli c ation in A as ω A = inf { τ : rk A n = O ((dim A n ) τ ) for all n > 1 } . 5 Obviously , 0 6 ω A 6 2. Note that this d efinition makes only sense if A conta ins algebras of arbitrarily big dimensions. I n th is case ω A > 1 since multipli cation in algebra is alwa ys faithful. This notion is very similar to the well-kno wn exp onent of matrix multiplic ation which will be denoted just by ω when th e ground field will b e clear. The only technical difference is th at the exp onent of matrix multiplica tion is defined relative to the square ro ot of the resp ective alg ebra dimension. In fact, it can b e easily seen t h at the regular exp onent of matrix m ultiplication eq u als doub le the rank -exp onent of matrix multiplication. W e ac knowledge that the introd uced rank-exp onent provides quite a crude estimate, since it even do es n ot indicate the grow th order of the bilinear complexity as a function of algebra dimension. F or example, if rk A n = O (dim A n ), t h en ω A = 1, bu t t h e opp osite statement must not hold: if ω A = 1 then the rank may p otentially b e sup erlinear, e.g. (dim A n ) · p olylog (dim A n ). On the other hand, there are no kno wn gen- eral upp er bound s t h at are tigh t enough for the rank -exp onent to b e too rough. One of the most f amous op en problems in computational li n- ear alg ebra and algebraic complexity theory is matrix m ultiplication, for whic h its exp onent (and th e rank exp onent) is only known to b e within 2 6 ω 6 2 . 376 [11]. 4 Structure of Group Algebras Here we introdu ce some basic concept s from the representa tion theory . F or the extensive treatmen t we refer to [27]. Let G b e a finite group and k b e a field. Then k [ G ] is semisimple if and only if c har k ∤ ♯G . Let G b e a fin ite group and k b e an algebraicall y closed field either of chara cteristic 0 or p ∤ ♯G . Then k [ G ] decomp oses into a direct pro duct of matrix algebras: k [ G ] ∼ = k n 1 × n 1 × · · · × k n t × n t , (1) where eac h matrix algebra is called irr e ducible r epr esentat ion of G ov er k , and t X τ =1 n 2 τ = ♯G. The numbers n 1 , . . . , n t are called the char acter de gr e es of G in k . If k is not algebraical ly closed but again of characteri stic either 0 or p ∤ ♯G , then k [ G ] ∼ = D n 1 × n 1 1 × · · · × D n t × n t t , (2) where D τ are all division algebras ov er k of dimensions d τ for 1 6 τ 6 t and t X τ =1 n 2 τ d τ = ♯G. Let k be a field of c haracteristic p and let G be a finite group of order np s , p ∤ n . Supp ose that a Sylow p -subgroup P ⊆ G is normal. Then J ( k [ G ]) is generated by J ( k [ P ]) (under the natural inclusion k [ P ] ⊆ k [ G ]) and dim J ( k [ G ]) = n ( p s − 1) . 6 According to the proposition 1, J ( k [ G ]) = rad k [ G ] and k [ G ] / rad k [ G ] is semisimple (see [26 ]). This implies k [ G ] /J ( k [ G ]) ∼ = D n 1 × n 1 1 × · · · × D n t × n t t , (3) where D τ again are all div ision algebras o ver k of dimension d τ for 1 6 τ 6 t and t X τ =1 n 2 τ d τ + dim J ( k [ G ]) = ♯G. (4) In case when S ylow p -subgroups of G are not normal the situation b e- comes more obscu re. How ever, it is known that J ( k [ G ]) contains all ideals generated b y J ( k [ H ]) where H is any normal p -subgroup of G . In partic- ular, this holds when H is the intersection of all the p -Sylow subgroups of G . 5 Bounds for t he Rank of Asso ciativ e Algebras and Complexit y of Matrix Multiplication One general lo wer boun d for the multiplicativ e (and therefore the bilin- ear) complexity of asso ciativ e algebras is due to A lder and Strassen. Theorem 2 ([1] ). L et A and B b e asso ci ative algebr as over a field k and let t ( A ) b e the numb er of maxim al twoside d ide als of A . Then C ( A × B ) > 2 dim A − t ( A ) + C ( B ) , (5) Algebras for which the Alder-St rassen b ound is tight (put B = { 0 } in (5)) are called al gebr as of mi nimal r ank . All such algebras ov er arbitrary fi elds w ere c haracterized by Bl¨ aser. Theorem 3 ([5] ). An algebr a A over an arbitr ary field k is an algebr a of minimal r ank iff A ∼ = C 1 × · · · × C s × k 2 × 2 × · · · × k 2 × 2 | {z } u times × B , (6) wher e C 1 , . . . , C s ar e lo c al al gebr as of mi ni mal r ank with dim( C σ / rad C σ ) > 2 , i.e., C σ ∼ = k [ X ] / ( p σ ( X ) d σ ) for some irr e ducible p olynomial p σ ( X ) with deg p σ > 2 , d σ > 1 , and ♯k > 2 dim C σ − 2 and B is a sup erb asic algebr a of mi nimal r ank; that is, ther e exist w 1 , . . . , w m ∈ rad B with w 2 i 6 = 0 and w i w j = 0 for i 6 = j such that rad B = L B + B w 1 B + · · · + B w m B = R B + B w 1 B + · · · + B w m B and ♯k > 2 N ( B ) − 2 . Any of the inte gers s, u , or m may b e zer o, and the factor B in (6) is optional. The next t wo lo wer b oun ds are du e to Bl¨ aser. 7 Theorem 4 ([3] ). L et A b e a finite dimensional algebr a over a field k , A/ rad A ∼ = A 1 × · · · × A t with A τ = D n τ × n τ τ for al l τ , wher e D τ is a k -divi sion algebr a. Assume that e ach factor A τ is nonc ommutative, that is, n τ > 2 or D τ is nonc ommutative. L et n = n 1 + · · · + n t . Then rk A > 5 2 dim A − 3 n. W e will show later how this can b e com bined with Theorem 2 for group algebras to obtain high low er b ounds in cases when some A τ are com- mutativ e. The nex t theorem gives a particularly go o d low er b ound for algebras with big radical and small n ilpoten ce index. Theorem 5 ([3] ). L et k b e a field and A b e a finite dimensional k - algebr a. F or al l m, n > 1 , the r ank of A i s b ounde d by rk A > dim A − d im((rad A ) n + m − 1 ) + dim((rad A ) m ) + dim((rad A ) n ) . (7) The follo wing fact is a simplified version of S ch¨ onhage’s τ -theorem. Theorem 6 ([24]). L et A = k n 1 × n 1 × · · · × k n t × n t , wher e n τ > 1 for at l e ast one τ and rk A 6 r . L et ω 0 b e a r o ot of the e quation n x 1 + · · · + n x t = r. Then the exp onent of matrix multiplic ation over k do es not exc e e d ω 0 . 6 Lo w er Bounds Let G = { G 1 , G 2 , . . . } b e a family of fi nite groups of unb ounded ord ers and let k be a field. W e will distinguish b etw een tw o different cases: 1. char k = 0 or char k = p and for an y i > 1 p ∤ ♯G i and 2. char k = p and for some i > 1 p | ♯G i . W e will call G in the first case a semisimple family of groups and in the second a mo dular family of groups. W e will start with the semisimple case. 6.1 Semisi mple Case W e will start with the case of algebraica lly closed k since all simple algebras over k are simply matrix algebras. Lemma 1. L et n 1 , . . . , n t > 0 and δ > 1 . Then t X τ =1 n τ 6 t 1 − 1 δ t X τ =1 n δ τ ! 1 δ . (8) 8 Pr o of. Let x 1 , . . . , x t , y 1 , . . . , y t b e complex numbers and a, b > 1 b e such that 1 a + 1 b = 1. Then, by H¨ older’s inequality t X τ =1 | x τ | | y τ | 6 t X τ =1 | x τ | a ! 1 a t X τ =1 | y τ | b ! 1 b . Choosing x τ = n τ and y τ = 1 for all τ , a = δ , and 1 b = 1 − 1 δ completes the p roof. ⊓ ⊔ Let G be a finite group and k be a field. W e introduce follo wing n otation: let t i ( G ) be the num b er of irreducible c haracter degrees of G ove r k equal to i . Let T i ( G ) = P ∞ j = i t j ( G ) b e the num ber of irreducible character degrees of G ov er k not less than i . Obviousl y , T i ( G ) > T j ( G ) , if i < j ; t i ( G ) = T i ( G ) − T i +1 ( G ); ♯G = ∞ X i =1 i 2 t i ( G ); t i ( G ) = 0 , if i > p ♯G − 1 . The last fol lo ws from the fact, t hat every group has a t l east tw o different irreducible representations. Note, t hat the number of maximal tw osided ideals of k [ G ] is exactly T 1 ( G ) = t , where t is the num b er of m ultiplicands in (1). Theorem 7. L et G b e a finite gr oup and k b e an algebr ai c al ly close d field of char acteristic ei ther 0 or p ∤ ♯G . L et t b e as in (1) . 1. If T 3 ( G ) = 0 then k [ G ] is of mini mal r ank and rk k [ G ] = 2 ♯G − t = t 1 ( G ) + 7 t 2 ( G ) . 2. If T 3 ( G ) > 0 then k [ G ] is not of minim al r ank then rk k [ G ] > 2 ♯G − t + max  5 2 T 7 ( G ) , 1  . 3. L et G = { G 1 , G 2 , . . . } b e a fam ily of finite gr oups, ♯G n < ♯G n +1 for al l n > 1 . Assume that the numb er of irr e ducible char acter de gr e es of G ∈ G over k i s o ( ♯G ) . 3 Then the fol lowing lower b ound holds: rk k [ G ] > 5 2 ♯G − o ( ♯G ) . Pr o of. Co nsider the decomp osition (1) for k [ G ]. Note, that the num b er t is exactly the number of maximal tw osided ideals of k [ G ]. Assume w.l.o.g . that n 1 6 · · · 6 n t and let A b e the direct pro du ct of all th e 3 By using this notation we mea n that for any constan t c > 0 there exists suc h N > 0 that if G ∈ G and ♯G > N then the num b er of irredu cible character d egrees of G o ver k is smaller than c · ♯G . 9 matrix algebras from (1) of order 1 or 2 and let B b e the remaining prod uct: k [ G ] = A × B . Note, that dim A = t 1 ( G ) + 4 t 2 ( G ) = T 1 ( G ) + 3 T 2 ( G ) − 4 T 3 ( G ) , (9) rk A = t 1 ( G ) + 7 t 2 ( G ) = 2 dim A − ( t 1 ( G ) + t 2 ( G )) . (10) (10) and the fa ct that A is of minimal rank fo llo w from Theorem 3. The num b er of maximal tw osided ideals in A is t 1 ( G ) + t 2 ( G ). 1. Let k [ G ] = A . Then T 3 ( G ) = 0, t = t 1 ( G ) + t 2 ( G ) and t h eorem follo ws from (10). 2. Let B b e nonempty . By Theorem 3, k [ G ] is n ot of minimal rank, therefore rk k [ G ] > 2 ♯G − t + 1. By (5) and the fact that A is of minimal rank rk k [ G ] = rk A × B = 2 dim A − ( T 1 ( G ) − T 3 ( G )) + rk B . The lo w er b ound follo ws from (5) and the upp er from the t rivial inequality rk A × B 6 rk A + rk B . Let B = B 1 × B 2 where B 1 conta ins all matrix algebras of (1) of order 6 6. The number of maximal tw osided ideals in B 1 is t 3 ( G ) + · · · + t 6 ( G ) = T 3 ( G ) − T 7 ( G ). Then, using (5) on ce again rk B > 2 dim B 1 − ( T 3 ( G ) − T 7 ( G )) + rk B 2 . Assume th at B 2 is not empty . Recall, that n 1 6 · · · 6 n t and there- fore n t − T 7 ( G )+1 > 7. F or B 2 w e can use Theorem 4: rk B 2 > 5 2 t X τ = t − T 7 ( G )+1 n 2 τ − 3 t X τ = t − T 7 ( G )+1 n τ = 2 dim B 2 + t X τ = t − T 7 ( G )+1  n τ  n τ 2 − 3  > 2 d im B 2 + 7 2 T 7 ( G ) . Gathering it all together, w e get rk k [ G ] > 2 dim A + 2 dim B 1 + 2 dim B 2 − T 1 ( G ) + 5 2 T 7 ( G ) = 2 ♯G − t + 5 2 T 7 ( G ) , whic h pro ves the second statement of th e theorem. 3. Let t = o ( ♯G ). Let k [ G ] = k t 1 ( G ) × C , C is ob viously n ot empt y , and dim C = n 2 t − T 2 ( G )+1 + · · · + n 2 t . By Alder-Strassen theorem rk k [ G ] = rk k t 1 ( G ) + rk C > t 1 ( G ) + 5 2 dim C − 3 t X τ = t − T 2 ( G )+1 n τ . By using Lemma 1 for dimensions of factors of C an d setting δ = 1 2 w e obtain t X τ = t − T 2 ( G )+1 n τ 6 p T 2 ( G ) dim C 6 p t♯G = o ( ♯G ) . 10 On the other hand, the num b er t 1 ( G ) of different irreducible rep- resen tations of G of dimension 1 does n ot exceed t and therefore is also o ( ♯G ), therefore, d im C = ♯G − t 1 ( G ) = ♯G − o ( ♯G ). Therefore, rk k [ G ] > 5 2 ♯G − o ( ♯G ) ⊓ ⊔ R emark 1. The lo w er b oun d in case 2 can b e improv ed furth er by em- plo ying the lo w er b ou n d d u e to Bl¨ aser rk k n × n > 2 n 2 + n − 2 for n > 3 [4]. How ever, the b est we can achiev e by now is to employ Alder-St rassen lo w er b oun ds for all multiplica nds in (1) ex cept for one (of the biggest dimension) and use 2 n 2 + n − 2 for the last: if n 1 6 · · · 6 n t and n t > 3 then rk k n 1 × n 1 × · · · × k n t × n t > 2 ♯G + n t − t − 1 . Corollary 1. L et k b e an algebr aic al ly close d field of char acteristic 0 . 1. L et S n b e the symmetric gr oup of or der n ! . Then rk k [ S n ] > 5 2 n ! − o ( n !) . 2. L et GL (2 , q ) b e the ge ner al line ar gr oup of nonsingular 2 × 2 -matric es over GF ( q ) . Then rk k [ GL (2 , q )] > 5 2 ♯GL (2 , q ) − o ( ♯GL (2 , q )) . 3. L et S L (2 , q ) b e the sp e cial li ne ar gr oup of 2 × 2 -matric es over GF ( q ) with determinant 1 . Then rk k [ S L (2 , q )] > 5 2 ♯S L (2 , q ) − o ( ♯ S L (2 , q )) . 4. L et p n b e the n th prime numb er. L et F p n , p n − 1 b e a F r ob enius gr oup of or der p n ( p n − 1) define d by { a, b : a p n = b p n − 1 = 1 , b − 1 ab = a u } , wher e u is an element of or der p n − 1 i n Z ∗ p n [17]. Then rk k [ F p n , p n − 1 ] > 5 2 p 2 n − o ( p 2 n ) . 5. L et p n b e t he n t h prime numb er and l et G n b e a non-ab eli an p n -gr oup with an ab eli an sub gr oup of index p n . Then rk k [ G n ] > 5 2 ♯G − o ( ♯G ) . Pr o of. 1. The statement follo ws from the fact th at the num b er of dif- feren t irreducible representations of S n o ver k equ als the num b er of p artitions of n [16] which asymptotically is e π √ 2 n 3 4 n √ 3 = o ( n !) [14], the latter can be observed ea sily from the w ell-kn o wn asymptotic of factorial: n ! ∼ √ 2 π n  n e  n . 2. [17] The num b er of elements in GL (2 , q ) equals q 4 − q 3 − q 2 + q > 3 8 q 4 , The num b er of different irreducible represen tations of GL (2 , q ) is q 2 − 1 = o ( q 4 ). 3. [9] The num b er of elements in S L (2 , q ) equ als q 3 − q > 3 4 q 3 . The num b er of differen t irreducible representations of S L (2 , q ) is q − 4 if q is o dd and q − 1 if q is a pow er of 2; b oth are o ( q 3 ). 11 4. [17] The n u mber of different irredu cible representations of F p n , p n − 1 is p n = o ( p 2 n ). 5. [17] L et ♯G n = p m n . The num b er of different irreducible rep resen ta- tions of G is p m − 1 n + p m − 2 n − p m − 3 n = p m n  1 p n + 1 p 2 n − 1 p 3 n  = o ( p m n ). ⊓ ⊔ Note, that if the Direct Su m Co njecture w ere true, t h en from (1) for the rank of multiplication in the group algebra k [ G ] for algebraically closed k wo uld immediately follo w rk k [ G ] = rk k n 1 × n 1 + · · · + rk k n t × n t . It turns out t hat an insignificantly weak er version of the correspond ing lo w er b ound can b e p ro ved ind ep endently of the v alidit y of the D irect Sum Conjecture. Theorem 8. L et G = { G 1 , G 2 , . . . } b e a f amily of finite g r oups and k b e an al gebr aic al ly close d field whose char acteristic do es not divi de any of the or ders of gr oups f r om G . L et f ( N ) b e a function that for e ach G ∈ G the dimension of the lar gest irr e ducibl e r epr esentat ion of G i s at le ast f ( ♯G ) . Then rk k [ G ] > f ( ♯G ) ω , wher e ω is the exp onent of matrix multipli c ation over k . L et t ( N ) b e a function such that for e ach G ∈ G the numb er of differ ent irr e ducible r epr esentations of G do es not exc e e d t ( ♯G ) . Then rk k [ G ] > ( ♯G ) ω 2 t ( ♯G ) ω 2 4 − ω 2 Pr o of. The first sta tement trivially fo llo ws from th e observ ation t hat for any alge bras A, B ov er one fi eld rk A × B > max { rk A, rk B } . Let k [ G ] hav e decomp osition according to (1). Consider th e follo wing equation n x 1 + · · · + n x t = rk k [ G ] . Let ω 0 b e a ro ot of this equation. Then by Sch¨ onhage’s τ -theorem ω 6 ω 0 . In other w ords, u sing the fact that all n τ > 1 n ω 1 + · · · + n ω t 6 rk k [ G ] . On the other hand, b y employing Lemma 1 rk k [ G ] > t X τ =1 n ω τ = t X τ =1 ( n 2 τ ) ω 2 > t 1 − ω 2 · t X τ =1 n 2 τ ! ω 2 > ( ♯G ) ω 2 t ( ♯G ) ω 2 4 − ω 2 . whic h pro ves the theorem. ⊓ ⊔ Corollary 2. 1. If the numb er of di ffer ent irr e ducible r epr esentations of gr oups in the family do es not gr ow “to o fast” then the exp onent of matrix multiplic ation is at most twi c e the r ank exp onent of the c or- r esp onding family of gr oup algebr as. M or e pr e cisely, if t ( N ) = o ( N ε ) for any ε > 0 then ω k [ G ] > ω 2 . 12 2. In the same setting, if ω > 2 , then the r ank of gr oup algebr as fr om the famil y describ e d ab ove is sup erli ne ar on their dimensions. 3. If ω > 2 and f ( N ) ≫ N 1 ω then the gr oup algebr as fr om the c orr e- sp onding family of gr oups have sup erline ar biline ar c omplexity. One pr omising f am ily of finite gr oups which c ould help to achieve ω = 2 in [9] has f ( N ) = N 1 2 − ε for some fixe d ε > 0 . It fol l ows, that in gener al one should lo ok for ε > 1 2 − 1 ω > 0 . 079 sinc e otherwise the lower b ound dep ends on ω and is not sup erline ar iff ω = 2 . 4. If t ( N ) ≪ N 2 ω then the biline ar c omplexity of the c orr esp onding gr oup algebr as is sup erline ar pr ovide d ω > 2 . In p articular, this holds if t ( N ) 6 N 0 . 841 . Corollary 3. L et k b e an algebr aic al ly close d field of char acteristic 0 . 1. L et { S n } n > 1 b e the family of symmetric gr oups, S n to b e of or der n ! . Then ω k [ S n ] = ω 2 . 2. L et { GL ( n, q ) } n > 1 , q fixe d, b e the famil y of gener al line ar gr oups of nonsingular n × n -matric es over GF ( q ) . Then ω k [ GL ( n, q )] = ω 2 . Pr o of. 1. F or the pro of refer to Coroll ary 1. 2. The order of GL ( n, q ) is N = n − 1 Y i =1  q n − q i  = q n 2 n − 1 Y i =1  1 − 1 q i  | {z } =: Q . Note that  1 − 1 q  n − 1 6 Q 6 1. GL ( n, q ) has an analytical irre- ducible representation of order d = n − 1 Y i =1  q i − 1  = n − 1 Y i =1 q i  1 − 1 q i  = q n ( n − 1) 2 Q, [13]. It follow s, that at least one irredu cible represen tation of has th e same order. No w th e corresponding matrix algebra has dimension d 2 = q n 2 − n Q 2 = N Q q n . W e will sho w no w that q n Q = o ( N ε ) for any ε > 0. This will complete the p roof since rk k [ GL ( n, q )] > d ω =  d 2  ω 2 > N (1 − ε ) ω 2 for all groups of size N > N 0 and ε > 0 where N 0 dep ends on th e choi ce of ε . q n Q 6 q n  1 − 1 q  n − 1 6 q 2 n − 1 . N ε > q εn 2  1 − 1 q  ε ( n − 1) > q εn 2 − εn . So N ε > q n Q if n > 2 ε + 1. ⊓ ⊔ 13 6.2 Mo dular Case Let k be now an algebraically closed fi eld of characteristic p and let G b e a finite group of order N = np d , where p ∤ n . W e will assume that G has the normal Sy lo w p -su bgroup H of order p d . In this case rad k [ G ] is generated by the augmentation ideal 4 of k [ H ] and dim rad k [ G ] = p d ( n − 1). W e will further b e concerned with the case of ab elian H , which is then a direct produ ct of cyclic p -groups: H = Z p t 1 × · · · × Z p t s , t 1 > · · · > t s , d = t 1 + · · · + t s . (11) W e will denote elements of H by h i 1 , ..., i s , 0 6 i σ < p t σ for al l 1 6 σ 6 s assuming h i 1 , ..., i s · h j 1 , ..., j s = h ( i 1 + j 1 ) mod p t 1 , ..., ( i s + j s ) mod p t s . Let r 1 = h 1 , 0 , 0 , ..., 0 − h 0 , 0 , 0 , ..., 0 , r 2 = h 0 , 1 , 0 , ..., 0 − h 0 , 0 , 0 , ..., 0 , . . . r s = h 0 , 0 , 0 , ..., 1 − h 0 , 0 , 0 , ..., 0 . The augmentation ideal of k [ H ] (and R = rad k [ G ]) is generated b y r 1 , . . . , r s . It is easy to see that r p t σ σ = 0 and the system of vectors n r i 1 1 · · · r i s s | i 1 + · · · + i s > 1 , 0 6 i σ < p t σ o is linearly indep endent. The system n r i 1 1 · · · r t s s | i 1 + · · · + i s > m, 0 6 i σ < p t σ o is also linearly indep end ent and generates R m , so dim R m = n ( p d − a m − 1 ) where a m − 1 = ♯  ( i 1 , . . . , i s ) | i 1 + · · · + i s 6 m − 1 , 0 6 i σ < p t σ  . Let ξ b e a discrete random v ariable. W e denote by E ξ the exp ectation of ξ , i.e. if ξ takes v alue a i ∈ R with p robabilit y p i > 0 for 1 6 i 6 n , P n i =1 p i = 1, then E ξ = P n i =1 a i p i . W e also denote by D ξ = E ( ξ − E ξ ) 2 the d ispersion of ξ . Theorem 9. L et G = { G 1 , G 2 , . . . } b e a f amily of gr oups and k b e a field of char acteristic p . L et G ∈ G and ♯G = N = np d , wher e p ∤ n . Assume that P = Z ( G ) 5 is the Sylow p -sub gr oup of G and the p ar ameter d is unb ounde d f or gr oups in G . L et p T b e the or der of biggest cycli c factor of P and p t b e the smal lest or der, and let s b e the total numb er of 4 The augmen tation ideal of a group algebra A with a group basis { e 1 , . . . , e n } is th e ideal generated by al l vecto rs P x i e i with P x i = 0. 5 Z ( G ) is the center of G , i.e. t h e set of elements of G that comm ute with all the elemen ts of G . 14 factors. Assume that f or any ε > 0 the differ enc e T − t < 1 2 log p εs for al l G ∈ G with ♯G > N 0 = N 0 ( ε ) . Then C ( k [ G ]) >  2 + 1 n  ♯G − o ( ♯G ) . Pr o of. F ollo wing pro of is based on ideas by Chok ay ev and generalizes similar result prove n in [7] for one special case of commutativ e group algebras. W e note, that since P is abelian, it is a fi nite product of cyclic p -groups: P = Z p t 1 × · · · × Z p t s where t 1 6 · · · 6 t s and the exp onent of P is p t s . Since it is o ( ♯P ), t h e parameter s is unboun ded among all groups from G . According to (7) C ( k [ G ]) > ♯G + n ( p d − a m − 1 ) + n ( p d − a m − 1 ) − n ( p d − a 2 m − 1 ) =  2 + a 2 m − 1 − 2 a m − 1 np d  ♯G. W e will sho w now that we ma y c ho ose m in such a w a y th at a 2 m p d → 1, a m p d → 0 when s → ∞ . Consider indices { i σ } s σ =1 as indep endent random v ariables with i σ taking v alue in [0 , p t σ − 1] with p robabilit y 1 p t σ for 1 6 σ 6 s . Then E i σ = p t σ − 1 2 , D i σ = p 2 t σ − 1 12 , and d enoting ξ s = i 1 + · · · + i s E ξ s = 1 2 s X σ =1 p t σ − s 2 , D ξ s = 1 12 s X σ =1 p 2 t σ − s 12 , while ξ s takes each v alue in [0 , P s σ =1 p t σ − s ] w ith probabili ty a m − a m − 1 p d . Now let m = 2 3 E ξ s b e a function of s . Then by Chebysho v ’s inequality a m − 1 p d = P ( ξ s 6 m − 1) 6 P ( | ξ s − E ξ s | > E ξ s − m + 1) 6 D ξ s ( E ξ s − m + 1) 2 6 3 sp 2 T 4 s 2 p 2 t = 3 p 2 T − 2 t 4 s − − − → s →∞ 0 , a 2 m − 1 p d = P ( ξ s 6 2 m − 1) > P ( | ξ s − E ξ s | 6 2 m − 1 − E ξ s ) > 1 − D ξ s (2 m − 1 − E ξ s ) 2 > 1 − 3 p 2 T − 2 t 4 s − − − → s →∞ 1 whic h pro ves the theorem. ⊓ ⊔ Corollary 4. F or any field k of char acteristic p and any family of gr oups { G 1 , G 2 , . . . } of gr owing dimensions ther e exists a c onstant N such that the gener ate d famil y of gr oup algebr as { k [ G 1 ] , k [ G 2 ] , . . . } do es not c on- tain algebr as of mini mal r ank of dimensions gr e ater th an N i f their Sylow p -sub gr oups c oi ncide wi th their c enters and c ontain gr owing numb er of cyclic factors of close or der. 15 7 Upp er Bounds As (1 ) and (2) indicate, complexity of multipli cation in group algebras is closely related to complexity of matrix multiplication. In particular, provided an effective a lgorithm for m ultiplication of square matrices, w e immediately obtain an effective algorithm for multipli cation in group algebras. Proposition 2. L et n 1 , . . . , n t > 0 and al pha > 1 . Then t X τ =1 n α τ 6 t X τ =1 n τ ! α . Pr o of. The statement foll ow s from the fact that x α is conv ex for x > 0 and α > 1. F or any pair of monotonically gro wing functions f ( n ) and g ( n ) we will write f ( n ) / g ( n ) if for every δ > 1 f ( n ) 6 O  ( g ( n )) δ  . Let G b e a finite group and k b e an algebraically closed field whose chara cteristic is either 0 or do es not divide ♯G . Now we are ready to introduce th e general u pp er b ound for the rank of k [ G ]. Theorem 10. L et G b e a gr oup and k b e an algebr ai c al ly close d field of char acteristic either 0 or c oprime wi th ♯G . Then rk k [ G ] / ( ♯G ) ω 2 , (12) wher e ω is the exp onent of matrix mul ti pl ic ation. Pr o of. Co nsider decomp osition (1) of k [ G ] into a direct pro duct of matrix algebras. It follo ws that rk k [ G ] 6 t X τ =1 rk k n τ × n τ . By defi nition of th e exp onent of matrix multiplication rk k n τ × n τ 6 L ( k n τ × n τ ) / n ω τ . Thus b y Prop osition 2 rk k [ G ] / t X τ =1 n ω τ = t X τ =1 ( n 2 τ ) ω 2 6 t X τ =1 n 2 τ ! ω 2 = ( ♯G ) ω 2 whic h completes the p roof. ⊓ ⊔ Lemma 2. L et G = { G 1 , G 2 , . . . } b e a famil y of finite gr oups and k b e an algebr aic al l y close d field of char acteristic either 0 or c oprime with e ach ♯G i . L et f ( N ) b e a function which satisfies fol lowing pr op erty: for every G ∈ G al l char acter de gr e es of G over k ar e less or e qual than f ( ♯G ) . Then for any G ∈ G rk k [ G ] / ♯ G · min h ( N )  h ( ♯G ) ω + f ( ♯G ) ω h ( ♯G ) 2  , (13) wher e ω is the exp onent of m atrix multiplic ation and the mini mum is taken over al l functions h ( N ) such that at le ast one irr e ducible char acter de gr e e of G is less or e qual than h ( ♯G ) . 16 Pr o of. Let n 1 > · · · > n t b e the irreducible character degrees of G ov er k . Let h ( N ) b e as defin ed . Let j ( N ) b e the num ber of n τ greater th an h ( N ). Note th at ♯G = t X τ =1 n 2 τ > j ( ♯G ) h ( ♯G ) 2 , thus j ( N ) 6 N h ( N ) 2 . It follow s that rk k [ G ] /   j ( ♯G ) f ( ♯G ) ω + t X τ = j ( ♯G )+1 n ω τ   6 ♯G f ( ♯G ) ω h ( ♯G ) 2 + ♯Gh ( ♯G ) ω . The last equation holds for any h ( N ) so it holds also for the one mini- mizing th e right side. ⊓ ⊔ Theorem 11. L et G = { G 1 , G 2 , . . . } b e a f amily of finite gr oups and k b e an algebr ai c al ly close d field of char acteristic either 0 or c oprime with or der of e ach G i . L et f ( N ) b e a function which satisfies f ol lowing pr op erty: for e ach G ∈ G al l char acter de gr e es of G over k ar e less or e qual than f ( ♯G ) . Then f or any G ∈ G rk k [ G ] / ♯Gf ( ♯G ) ω − 2+ 4 ω +2 6 ♯Gf ( ♯G ) ω − 1 , (14) wher e ω is the exp onent of matrix mul ti pl ic ation. Pr o of. It is a w ell-k now n fact th at ev ery group has at least one (t riv ial) one-dimensional rep resen tation. So w e can choose for h ( N ) in Lemma 2 any function which is less than f ( N ). The result of the theorem follow s by c hoosing h ( N ) = f ( N ) 1 − 2 ω +2 . ⊓ ⊔ Corollary 5. 1. If f ( N ) = O (1) then rk k [ G i ] = O ( N ) . 2. If for any ε > 0 f ( N ) = o ( N ε ) then ω k [ G ] = 1 . R emark 2. 1. Note, th at h ( N ) =  2 ω  1 ω +2 f ( N ) 1 − ω ω +2 minimizes the righ t side of (13). 2. The upp er b ound giv en by (14) is b etter than the one given by (12) if f ( N ) = o  N 1 2 − 2 ω 2  . According to th e b est known upp er b ound ω < 2 . 376 [11], currently (14) b eats (12) if f ( N ) = o ( N 0 . 1457 ). Let k no w b e an arbitrary field of characteristic 0 and G be a finite group. By defi n ition of prime field, Q ⊆ k is the prime subfield of k . Let K ⊇ k b e an algebraica lly closed exten sion of k . It is k now n (see [18, Theorem 11.4, Chapter XVI I I]) that every rep resentation of G o ver K is d efi nable o ver Q ( ζ m ) where m is exp onent of G and ζ m is a primitive m -th ro ot of unity . Therefore, it is defin ab le ov er k ( ζ m ) (if k do es not already contain ζ m ). Now consider any irredu cible represen tation of G ov er k . It is a simple k [ G ]-mo du le by Maschk e’s Theorem [18, Theorem 1.2, Chapter XVI I I]. Therefore, it is isomorphic to D n × n where D is a k -division algebra. ζ m is algebraic o ver D since it is algebraic o ver k ⊆ D and D ∼ = D ′ ⊆ k ( ζ m ). The latter holds since there are no simple irreducible representations of G ov er k ( ζ m ) other than those isomorphic to matrix al gebras over k ( ζ m ). Thus, D is a subalgebra of k ( ζ m ), or D ∼ = k ( ζ ℓ ) for some ℓ | m . 17 Theorem 12. L et G = { G 1 , G 2 , . . . } b e a family of finite gr oups and k b e an arbitr ary field of char acteristic 0 . Then for any G ∈ G rk k [ G ] / ( ♯G ) ω 2 , wher e ω is again the exp onent of matrix m ul tiplic ation. Pr o of. Since k [ G ] is semisimple, (2) holds. As mentioned ab o ve, D τ is actually an exten sion field of k , thus for all τ rk D τ 6 2 d τ − 1 since it can b e implemented via p olyn omial multiplication ov er k and k is infinite. W e hav e rk k [ G ] / t X τ =1 n ω τ (2 d τ − 1) < 2 t X τ =1 n ω τ d τ = 2 t X τ =1  n 2 τ d 2 ω τ  ω 2 6 2 t X τ =1 n 2 τ d 2 ω τ ! ω 2 6 2 t X τ =1 n 2 τ d τ ! ω 2 = 2( ♯G ) ω 2 since ω > 2. ⊓ ⊔ R emark 3. Statement of th eorem 12 rema ins true whenever the division algebras app ear in side simple irreducible representations of groups h a ve linear rank . Thus, 1. Theorem 12 holds also when k is finite. It is kn own that any finite division algebra is an ex tension field of k , by W edderburn’s Lit- tle Theorem [19, Theorem 2.55], therefore its rank is linear due to Chudno vskys’ algorithm, cf. [8] or [25]. 2. It also holds for real cl osed fields s ince all division algebras ov er such fields hav e b ounded dimension (in fact, it can b e only 1, 2, 4, or 8) [12]. 8 Conclusion Noncommutativ e group algebras app ear to b e closely connected w ith the matrix algebra. S tudying the p roblem of complexity of multiplicatio n in group algebras ma y give us new algebraic insight into this classical problem of compu t er algebra and algebraic co mplexity theory . There are numerous op en problems related to group algebras. W e mention here only some of them. 1. It could b e p ossible t o obtain a general u pp er b ound not dep ending on th e matrix representations for th e rank of group algebras based on the group structure that will b e better than upp er b oun d s given by Theorems 10, 11, and 12. In th is case it could improv e the u pp er b ound for matrix multiplica tion. 2. W e would lik e t o extend Theorem 12 for fields of arbitrary charac- teristic that does not divide any of the g roup orders f rom the fa mily under consideration. 18 3. The radical of a group algebra in the mo dular case is tightl y re- lated to Sy lo w p -groups. These groups are w ell-studied, although their structure may v ary v ery strongly . I t is k now n th at the rank of commutativ e group algebras with n ontrivial radical is still lin- ear, so it do es not affect the order of the complexity . On the other hand, a comm utative group algebra ov er algebraica lly closed field of c haracteristic p is of minimal rank iff its S y lo w p -group is cy clic. 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