Abstract:
Let $\mathfrak X$ 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 $\mathfrak X$-subgroup if there exists an isomorpic embedding $\phi\colon G\hookrightarrow G^*$ of the group $G$ into some finite group $G^*$ under which $G^\phi$ is subnormal in $G^*$ and $H^\phi=K\cap G^\phi$ for some maximal $\mathfrak X$-subgroup $K$ of $G^*$. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal $\mathfrak X$-subgroups are conjugate in a finite group $G$ in which all maximal $\mathfrak X$-subgroups are conjugate? This question strengthens Wielandt's known problem of closedness for the class of $\mathscr D_\pi$-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.
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