Conjugacy of Maximal and Submaximal ????-Subgroups

Abstract

Let ???? be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal ????-subgroup if there exists an isomorpic embedding ϕ: G ↪ G^* of the group G into some finite group G^* under which G^ϕ is subnormal in G^* and H^ϕ = K ∩G^ϕ for some maximal ????-subgroup K of G^*. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal ????-subgroups are conjugate in a finite group G in which all maximal ????-subgroups are conjugate? This question strengthens Wielandt’s known problem of closedness for the class of -groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where G is a simple group.