Generation and random generation: From simple groups to maximal subgroups

Abstract

Abstract. Let G be a finite group and let d(G) be the minimal number of generators for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We prove that d(H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.

Describing finite groups by short first-order sentences

“…6. Let A be a finite abelian group, X a set and let V ≤ A X be a subgroup of rank d. There exists a set Y ⊆ X of size at most d · λ(|A|) such that for all g ∈ V , g ↾ Y = 0 implies g = 0.…”

“…6. The class of finite solvable groups, and more generally of groups without a composition factor of type 2 G 2 , is strongly log 3 -compressible.…”

Generating sets of finite groups

“…, T r are isomorphic nonabelian simple groups; G/N has order rm for some m dividing |Out(T 1 )|, and induces a cyclic permutation of the factors. We shall show that Aut * (G) is trivial for groups of type (1), and is equal to Aut(G) for groups of type (3) and (4). Furthermore, we shall show that in type (1) there is a spectacularly large gap between Aut(Γ(G)) and Aut(Γ w (G)), whilst in type (2) and (3) we find that Aut * (G) = Aut(Γ w (G)).…”

“…Burness, Liebeck and Shalev prove (see [4,Theorem 7]) that the point stabiliser of a d-generated finite primitive permutation group can be generated by d + 4 elements. Hence if G is a finite group, thenm(G) ≤ d(G) + 4 and our first claim follows from Theorem 2.10.…”

“…In Theorem 3. 4 we characterise the soluble groups G on which these two relations coincide; it would be very interesting to determine for which insoluble groups they are equal.…”