Asymptotic Results on Modular Representations of Symmetric Groups and Almost Simple Modular Group Algebras

Abstract

In fact, this is a question about infinite simple groups because it is easy for G finite, and because a non-trivial normal subgroup gives rise to a non-trivial ideal different from the augmentation ideal. Also note that the problem reduces easily to the question on when the augmentation ideal is simple as a ring. The first interesting class of groups with almost simple FG was discovered in [3]. This class is rather exotic and contains groups like the universal Hall group and algebraically closed groups. For locally finite groups G a substantial progress was achieved recently by using representation theory of finite groups, see [18, 19, 16], and others. The representation theory approach transforms the problems on ideals to certain problems on asymptotic behavior of representations of finite groups, which are often of independent interest. The method is more effective over fields of characteristic zero as the theory of ordinary representations is much better elaborated than the modular theory. This paper is devoted to the modular aspect of the theory. One of our main results deals with the case where G is a direct limit of alternating groups and F is any field of characteristic p > 2. We note that the case where charF = 0 was considered in [18]. Let N be the set of natural numbers. Denote by Alt(Ω) and Sym(Ω) (or, simply, An and Σn) the alternating and symmetric groups, respectively, on a set Ω with |Ω| = n. Let Alt(Ω1) ⊂ Alt(Ω2) ⊂ · · · ⊂ Alt(Ωi) ⊂ . . . (1)