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## Complex group algebras of finite groups: Brauer's

Problem 1

### Alexander Moretó

**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.

*Copyright 2005 American Mathematical Society
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#### Article Info

- ERA Amer. Math. Soc.
**11** (2005), pp. 34-39
- Publisher Identifier: S 1079-6762(05)00144-7
- 1991
*Mathematics Subject Classification*. Primary 20C15
- Received by editors October 12, 2004
- Posted on May 10, 2005
- Communicated by David J. Benson
- Comments (When Available)

**Alexander Moretó**

Departament d'Àlgebra,
Universitat de València,
46100 Burjassot, València, SPAIN

*E-mail address:* `Alexander.Moreto@uv.es`

Research supported by the Basque Government, the Spanish
Ministerio de Ciencia y Tecnología, grant BFM2001-0180, and
the FEDER

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