Pronormality of Hall subgroups in their normal closure

E. P. Vdovin ; M. N. Nesterov ; D. O. Revin

Algebra i logika, Tome 56 (2017) no. 6, pp. 682-690

Résumé

It is known that for any set $\pi$ of prime numbers, the following assertions are equivalent: (1) in any finite group, $\pi$-Hall subgroups are conjugate; (2) in any finite group, $\pi$-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, $\pi$-Hall subgroups are pronormal in their normal closure. Previously [Unsolved Problems in Group Theory, The Kourovka Notebook, 18th edn., Institute of Mathematics SO RAN, Novosibirsk (2014), Quest. 18.32], the question was posed whether it is true that in a finite group, $\pi$-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [Sib. El. Mat. Izv., 12 (2015), 1032–1038] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set $\pi$. The fact that there exist examples of finite sets $\pi$ and finite groups $G$ such that $G$ contains more than one conjugacy class of $\pi$-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for $\pi$ is unessential for (1), (2), and (3) to be equivalent.

Détail

Keywords: $\pi$-Hall subgroup, normal closure, pronormal subgroup.