Complex group algebras of finite groups: Brauer’s Problem 1

Moretó, Alexander

doi:10.1090/S1079-6762-05-00144-7

Complex group algebras of finite groups: Brauer’s Problem 1

Author: Alexander Moretó
Journal: Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 34-39
MSC (1991): Primary 20C15
DOI: https://doi.org/10.1090/S1079-6762-05-00144-7
Published electronically: May 10, 2005
MathSciNet review: 2150942
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Abstract: Brauer’s Problem 1 asks the following: what are the possible complex group algebras of finite groups? It seems that with the present knowledge of representation theory it is not possible to settle this question. The goal of this paper is to announce a partial solution to this problem. We conjecture that if the complex group algebra of a finite group does not have more than a fixed number $m$ of isomorphic summands, then its dimension is bounded in terms of $m$. We prove that this is true for every finite group if it is true for the symmetric groups.

László Babai and László Pyber, Permutation groups without exponentially many orbits on the power set, J. Combin. Theory Ser. A 66 (1994), no. 1, 160–168. MR 1273297, DOI https://doi.org/10.1016/0097-3165%2894%2990056-6
Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056
Paul Erdős, S. W. Graham, Aleksandar Ivić, and Carl Pomerance, On the number of divisors of $n!$, Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) Progr. Math., vol. 138, Birkhäuser Boston, Boston, MA, 1996, pp. 337–355. MR 1399347
Eliana Farias e Soares, Big primes and character values for solvable groups, J. Algebra 100 (1986), no. 2, 305–324. MR 840579, DOI https://doi.org/10.1016/0021-8693%2886%2990079-7
J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric groups, Canad. J. Math. 6 (1954), 316–324. MR 62127, DOI https://doi.org/10.4153/cjm-1954-030-1
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
Bertram Huppert, Research in representation theory at Mainz (1984–1990), Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 17–36. MR 1112156
Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin-New York, 1982. AMD, 44. MR 650245
I. Martin Isaacs, Character theory of finite groups, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423 (57 #417)]. MR 1280461
A. Jaikin-Zapirain, On the number of conjugacy classes in finite $p$-groups, J. London Math. Soc. (2) 68 (2003), no. 3, 699–711. MR 2009445, DOI https://doi.org/10.1112/S002461070300454X
Thomas Michael Keller, Orbits in finite group actions, Groups St. Andrews 2001 in Oxford. Vol. II, London Math. Soc. Lecture Note Ser., vol. 305, Cambridge Univ. Press, Cambridge, 2003, pp. 306–331. MR 2051537, DOI https://doi.org/10.1017/CBO9780511542787.003

lish M. Liebeck, A. Shalev, Character degrees and random walks in finite groups of Lie type, preprint.

Alexander Moretó and Josu Sangroniz, On the number of conjugacy classes of zeros of characters, Israel J. Math. 142 (2004), 163–187. MR 2085714, DOI https://doi.org/10.1007/BF02771531