The $D_\pi$-Property in a Class of Finite Groups

A finite group $G$ is a $D_\pi$-group for some set $\pi$ of primes if maximal $\pi$-subgroups of $G$ are all conjugate. Assume that every non-Abelian composition factor of the $D_\pi$-group $G$ is isomorphic either to an alternating group, or to one of the sporadic groups, or to a simple group of Lie type over a field whose characteristic belongs to $\pi$. We prove that an extension of $G$ by an arbitrary $D_\pi$-group and every normal subgroup of $G$ are $D_\pi$-groups. This gives partial answers to Questions 3.62 and 13.33 in the “Kourovka Notebook”. Also, we describe all $D_\pi$-groups whose composition factors are isomorphic to alternating, sporadic, and Lie-type groups whose characteristics belong to $\pi$. And bring to a close the description of Hall subgroups in sporadic groups, initiated by F. Gross.