On the pronormality of Hall subgroups

TL;DR

This paper proves that in $C_$-groups, $$-Hall subgroups are pronormal and that this property is inherited by overgroups, providing an affirmative answer to a problem from the Kourovka notebook, with counterexamples included.

Contribution

It establishes the pronormality of $$-Hall subgroups in $C_$-groups and shows this property is inherited by overgroups, solving a longstanding problem.

Findings

Proves pronormality of $$-Hall subgroups in $C_$-groups.

Shows inheritance of $C_$ property by overgroups of $$-Hall subgroups.

Provides counterexamples of non-pronormal Hall subgroups in general.

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.