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 $$\alpha (G)$$ α ( G ) be the number of conjugacy classes of core-free subgroups H of G with $$b(G,H) \ge 3$$ b ( G , H ) ≥ 3 . We say that G is a strongly base-two group if $$\alpha (G) \le 1$$ α ( G ) ≤ 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.