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 isomorphic embedding ϕ : G ↪ G^* of 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^*. In the case where ???? coincides with the class of all π-groups for some set π of prime numbers, submaximal ????-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal ????- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal ????-subgroups are conjugate in a finite group G in which all maximal ????-subgroups are conjugate?