LAUSR

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 Gϕ is subnormal in G* and Hϕ = K ∩Gϕ 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.