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2 papers
Pronormality of Hall subgroups in their normal closure
E. P. Vdovinab,
M. N. Nesterovab,
D. O. Revinab a Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia
Abstract:
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.
Keywords:
$\pi$-Hall subgroup, normal closure, pronormal subgroup.
UDC:
512.542 Received: 18.04.2017
DOI:
10.17377/alglog.2017.56.603