On the pronormality of Hall subgroups

Abstract

Fix a set of primes $\pi$. A finite group is said to satisfy $C_{\pi }$ or, in other words, to be a $C_{\pi }$-group, if it possesses exactly one class of conjugate $\pi$-Hall subgroups. The pronormality of $\pi$-Hall subgroups in $C_{\pi }$-groups is proven, or, equivalently, we prove that $C_{\pi }$ is inherited by overgroups of $\pi$-Hall subgroups. Thus an affirmative solution to Problem 17.44(a) from the "Kourovka notebook" is obtained. We also provide an example, showing that Hall subgroups in finite groups are not pronormal in general.