ON STRONGLY BASE-TWO FINITE GROUPS WITH TRIVIAL FRATTINI SUBGROUP: CONJUGACY CLASSES AND CORE-FREE SUBGROUP

Abstract

This paper explores finite groups G with a focus on those that are strongly base-two and possess a trivial Frattini subgroup. The concept of base size, denoted by b(G, H), for the action of G on core-free subgroups H, plays acrucial role. The paper investigates the number of conjugacy classes of core-free subgroups with base size exceeding 3, denoted by α(G). A group is considered strongly base-two if α(G) ≤ 1, indicating that nearly all faithful transitive permutation representations of G exhibit a base size of 2. The study delves into the characterization of such groups, shedding light on their properties and structures.