Conjugacy of maximal and submaximal X-subgroups
Let X 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 X-subgroup if there exists an isomorpic embedding ф : G - > G* of the group G into some finite group G* under which is subnormal in G* and Н<sup>ф</sup> = K П for some maximal X-subgroup K of G*. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal X-subgroups are conjugate in a finite group G in which all maximal X-subgroups are conjugate? This question strengthens Wielandt's known problem of closedness for the class of D<sub>n</sub>-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.