On the probability of generating finite groups with a unique minimal normal subgroup

Abstract

Assume that a finite group G has a unique minimal normal subgroup, say N . We prove that if the order of N is large enough then the following is true: If d randomly chosen elements generate G modulo N , then these elements almost certainly generate G itself.

Generating minimally transitive permutation groups

“…Furthermore, it is shown in the proof of the main theorem in [9], that |∆ i+1 |/|∆ i | = |N i+1 : H i+1 | is greater than 1 for 0 ≤ i ≤ k − 2, and also for i = k − 1 if N is abelian. Note also that G/ Soc (G) ∼ = L/M is m-generated, by the previous paragraph; thus, L is m-generated (see [10]).…”

A reduction theorem for nonsolvable finite groups

“…Assume our claim true for groups of nonsolvable length at most n − 1 and choose H any n-rarefied group. It is harmless to assume that Φ(H) = 1, since any set of elements generating H modulo Φ(H), also generates H. Thus S 1 (H) is the unique minimal normal subgroup of H, and we can use the main result of [19] to see that the minimal number of generators of H/S 1 (H) is the same as the minimal number of generators of H. The inductive hypothesis yelds that H is 2-generated and Theorem 1.1 completes the proof.…”

Closed normal subgroups of free pro-S-groups of finite rank

“…Notice that LðS; kÞ is a monolithic group, socðLðS; kÞÞ G S l kÀ1 and if k > 1 then LðS; kÞ=socðLðS; kÞÞ G LðS; k À 1Þ. It follows from the classification of the finite simple groups that dðSÞ ¼ 2 and the main result of [9] implies that dðLðS; kÞÞ ¼ dðLðS; k À 1ÞÞ ¼ dðSÞ ¼ 2. The following holds.…”