Strongly Base-Two Groups.

Abstract

Let G be a finite group, let H be a core-free subgroup and let b(G, H) denote the base size for the action of G on G/H. Let $\alpha \left(G\right)$ be the number of conjugacy classes of core-free subgroups H of G with $b(G,H)\ge 3$. We say that G is a strongly base-two group if $\alpha \left(G\right)\le 1$, which means that almost every faithful transitive permutation representation of G has base size 2. In this paper, we study the strongly base-two finite groups with trivial Frattini subgroup.