22 - Some remarks on the structure of finite subgroups of simple algebraic groups

Summary

This paper is a survey of some results by the author in the study of the subgroup structure of the finite simple groups of Lie type. Throughout the paper G is a simple algebraic group; if G is denned over a finite field F_q then σ is some Steinberg endomorphism of G. We shall omit index G in notations like N_G(X), C_G(X).

Subgroups of simple groups of exceptional type.

The first result of the paper is a reduction theorem for the maximal subgroups of finite exceptional groups similar to the well-known result of M. Aschbacher for finite classical groups. In the case of classical groups Theorem 1 doesn't give any new information, but it may be useful in the study of simple groups of exceptional type. Another and more explicit version of a reduction theorem was obtained recently by M. W. Liebeck and G. M. Seitz.

Theorem 1Let G be defined over the finite field F_q, G_σ ≅ G(F_q) and. Let G₀ ≤ G₁ ≤ Aut G₀and let M be a subgroup in G₁. Then one of the following statements is valid:

1. (a) for some proper connected nontrivial σ-invariant subgroup H ≤ G;

2. (b) M is an almost simple group, i.e. S ≤ M ≤ Aut S for some simple group S;

3. (c) M ≤ N_G(J) for some Jordan subgroup J in G;

4. (d) G is of type E₈, charF_q = p > 5 and M ≤ NGl(X), where X ≅ Alt₅ × Alt₆.