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논문에서는 임의의 유한 군 G를 고려하고, G의 군 행렬 XG가 정의되며, 이로부터 군 결정인자(group determinant) ΘG가 유도된다. ΘG는 Frobenius가 1896년에 소개한 특수한 형태의 양의 고윳값을 가진 행렬이다.
논문에서는 Brauer가 제기한 문제에 답하기 위해 'k-특성' (k-character) 이라는 새로운 기법을 도입한다. k-특성은 각 군의 요소 g에 대해 χ(g)의 k차 적분과 같은 특성을 가지는 함수이다. 특히 1-특성(χ_1), 2-특성(χ_2), 3-특성(χ_3)가 정의된다.
논문에서는, 그룹을 결정하는 데 유용한 새로운 특성인 3-특성을 도입하고, 이 특성은 군을 완전히 구분한다는 결과를 얻는다. 또한, 이 연구는 Frobenius-Sibley 정리와 Formanek-Sibley 정리를 확장하여, k-특성(k-character)들이 군을 결정하는 데 충분하다는 것을 보여준다.
한글 요약 끝
영문 요약:
The paper, published in 1992 by H.-J. Hoehnke and K.W. Johnson, introduces a new method called 'k-character' to answer Brauer's question on what extra information can be added to the ordinary character table of a group to completely determine it.
The k- characteristic is defined as a function that maps each element g in the group to χ(g) integrated up to order k. Specifically, the 1-character (χ_1), 2-character (χ_2), and 3-character (χ_3) are defined.
The paper shows that the 3-character determines the group uniquely and that the knowledge of the k-characters for small values of k can determine a group up to isomorphism.
This research also extends the Frobenius-Sibley theorem and Formanek-Sibley theorem, showing that the k-characters are sufficient to determine a group completely.
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arXiv:math/9210219v1 [math.GR] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 243-245THE 1-, 2-, AND 3-CHARACTERS DETERMINE A GROUPH.-J. Hoehnke and K. W. JohnsonAbstract.
A set of invariants for a finite group is described. These arise naturallyfrom Frobenius’ early work on the group determinant and provide an answer to aquestion of Brauer.
Whereas it is well known that the ordinary character table ofa group does not determine the group uniquely, it is a consequence of the resultspresented here that a group is determined uniquely by its “3-character” table.1. IntroductionGiven a finite group G of order n its group matrix XG is defined as follows.
Letxe, xg2, . .
. , xgn be variables indexed by the elements of G. The n × n matrix XG isdefined to be the matrix whose (i, j)th entry is xgig−1j .
The group determinant ΘGof G is det(XG).Although the group determinant first appeared in Frobenius’ 1896 paper [Fr1] inwhich he introduced characters for arbitrary groups, the roots of the work go backto the discussions of Gauss on the composition of equivalence classes of quadraticforms. There is a good discussion of the background to Frobenius’ work in [Ha1,Ha2].
A natural question arises as to whether nonisomorphic groups necessarilyhave distinct group determinants. This was posed by one of the authors in 1986and was answered by Formanek and Sibley [FS] in 1990, the positive answer beingsomewhat surprising.
In fact there were already two early remarks in the literatureapart from the work of Gauss quoted above that pointed in this direction. Both ofthese were described in apostrophies as strange (“merkw¨urdig”) by their authorsand were disregarded for a long time within the development of algebra.
The firstwas by Frobenius in [Fr2] on matrix transformations and the second occurred in [B]on the subject of maximal orders of quaternion algebras. Today these phenomenaare subsumed under the thesis “from norm invariance to constructive structuretheory and (noncommutative) arithmetics.”A further look at Frobenius’ early work reveals functions χ(k): Gk →C definedbelow in the case where k ≤3 that appeared in his algorithm to calculate the factorof a group determinant that corresponds to an irreducible character χ.
Some of1991 Mathematics Subject Classification. Primary 15A15, 16A26, 16H05, 16K20, 16S99, 20B05,20B40, 20C15.Received by the editors March 20, 1991 and, in revised form, February 20, 1992c⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1
2H.-J. HOEHNKE AND K. W. JOHNSONthe properties of these “k-characters” have been explored in [J] where it is shownthat the k-characters corresponding to distinct irreducible characters χi, χj areorthogonal in the sense thatXg∈Gkχ(k)i(g)χ(k)j (g) = 0.There is also given in [J] a 2-character table of a group G that consists of the 2-characters corresponding to the irreducible representations of degrees greater than1 together with “degenerate” characters.One of the questions raised by Brauer in [Br] is that of determining what extrainformation may be added to the (ordinary) character table to completely determinea group.
As a consequence of Frobenius’ work and the Formanek-Sibley result, theknowledge of the k-characters χ(k) of a group for all k and all irreducible charactersχ is sufficient to determine a group. Note that if k > deg(χ) then χ(k) = 0.
Thek-characters thus provide an answer to Brauer’s question, but since the amount ofwork involved in calculating the k-characters for large k is prohibitively large, itbecomes interesting to examine the question of the extent to which the knowledgeof the k-characters for small values of k determine a group.2. 3-charactersThe authors would like to announce the following result.Theorem.
Let G be a finite group, with irreducible characters χ1, . .
. , χr over analgebraically closed field K with char K ̸= 2 and char K ∤|G|.
Then G is determinedup to isomorphism by the χ(k)j , j = 1, . .
. , r, k = 1, 2, 3.Explicit definitions of the 1-, 2-, and 3-characters corresponding to a characterχ are as follows:χ1(g) = χ(g),χ2(g, h) = χ(g)χ(h) −χ(gh),χ3(g, h, m) = χ(g)χ(h)χ(m) −χ(g)χ(hm) −χ(h)χ(gm)−χ(m)χ(gh) + χ(ghm) + χ(gmh).The proof of the theorem depends on results in [Ho1, Ho2]; the following is abrief outline:(i) To each irreducible character χ of a finite group G there is associated a factorϕχ of ΘG.
(ii) Any factor ϕ of ΘG is a norm-type form. In particular, ϕ(xy) = ϕ(x)ϕ(y)where x and y are generic elements of the group algebra of G.(iii) Let A be a finite-dimensional algebra over the field K, and let {ω1, .
. .
, ωn}be a basis. Define the structure constants {wkij} byωiωj =Xwkijωk.Let N be a norm-type form on A, withN(λ −x) = λm −s1(x)λm−1 + · · · + (−1)msm(x)
THE 1-, 2-, AND 3-CHARACTERS DETERMINE A GROUP3where λ is an indeterminate and x is a generic element,x = x1ω1 + · · · + xnωn.If the discriminant of N is nonzero (which is true for example when N is the groupdeterminant) it follows that from the knowledge of s1(x), s2(x), and s3(x) the“symmetrised” structure constantswk(ij) = wkij + wkjican be determined. (iv) By [Bo, §4, Exercise 26] if f: G →H is a bijection between finite groupssuch that f(gh) is either f(g)f(h) or f(h)f(g), then f is either an isomorphism oran anti-isomorphism, and hence G and H are isomorphic.
(v) It follows from (iii) and (iv) that the 3-character of the regular representation(which is essentially the same as s3(x) above when N = ΘG) determines G.(vi) The 3-character of the regular representation of G can be formed from the1-, 2-, and 3-characters associated to the irreducible representations of G.This provides, in some sense, a more satisfactory set of invariants for a finitegroup.It has also been possible to give a constructive proof of the Formanek-Sibley theorem using the ideas in the proof of the above theorem.Another set of invariants for finite groups has been given by Roitman in [Ro].These invariants consist of an apparently infinite set of integers defined for pairs ofintegers n, k that are calculated as the coefficient of the identity in the kth powersof certain elements of ZG ⊗· · · ⊗ZG (n factors), identifying this ring with ZGn.We refer the reader to [Ro] for the details, which are of a technical nature. The 1-,2-, and 3-characters associated to the irreducible representations of the group seemto be much more accessible, and moreover, a 3-character table can be constructedthat has convenient orthogonality properties.
The question of whether the 1- and2-characters alone determine a group is addressed in [JS] where it is shown thatthere exist nonisomorphic pairs of groups with the same 2-character table, i.e., withthe same 1- and 2-characters.References[B]H. Brandt, Idealtheorie in Quaternionenalgebren, Math. Ann.
99 (1928), 1–29.[Bo]N. Bourbaki, Alg`ebre.
I, Paris, 1970.[Br]R. Brauer, Representations of finite groups, Lectures in Modern Mathematics (T. L.
Saaty,ed. ), Wile, 1963, pp.
133–175.[FS]E. Formanek and D. Sibley, The group determinant determines the group, Proc.
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112 (1991), 649–656.[Fr1]G. Frobenius, ¨Uber die Primfaktoren der Gruppendeterminante, S’ber.
Akad. Wiss.
Berlin(1896), 1343–1382. [Fr2], ¨Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen, S’ber.Akad.
Wiss. Berlin (1898), 944–1015.
[Ha1] T. Hawkins, The origins of the theory of group characters, Arch. Hist.
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[Ha2], New light on Frobenius’ creation of the theory of group characters, Arch. Hist.Exact Sci.
12 (1974), 217–243. [Ho1] H.-J.
Hoehnke,¨Uber komponierbare Formen und konkordante hyperkomplexe Gr¨ossen,Math. Z.
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4H.-J. HOEHNKE AND K. W. JOHNSON[J]K. W. Johnson, On the group determinant, Math.
Proc. Cambridge Philos.
Soc. 109 (1991),299–311.[JS]K.
W. Johnson and Surinder Sehgal, The 2-character table does not determine a group,preprint.[Ro]M. Roitman, A complete set of invariants for finite groups, Adv.
in Math. 41 (1981),301–311.Mendelstr.
4, 0-1100 Berlin, Germany and Brandenburgische Landesuniversit¨at,Fachbereich Mathematik, AM Neuen Palais, 0-1571 Potsdam-Sanssouci, GermanyDepartment of Mathematics, The Pennsylvania State University Abington, Penn-sylvania 19001E-mail address: kwj1 @ psuvm.bitnet
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