On pronormal subgroups in finite simple groups
type: Item
instance of: scholarly article
title: On pronormal subgroups in finite simple groups
zbMATH Open document ID: 1418.20006
DOI: 10.1134/S1064562418060029
author: Natal'ya Vladimirovna Maslova
author: Anatoli Kondrat'ev
author: D. O. Revin
published in: Doklady Mathematics
publication date: 2019-01-15T00:00:00Z
review text: A subgroup $H$ of a group $G$ is called pronormal if for every $g\in G$, the groups $H$ and $H^g$ are conjugate in the subgroup they generate. In particular, normal subgroups and maximal subgroups are pronormal, as are Hall subgroups of solvable groups. The conjecture that subgroups of odd index in simple groups are pronormal turns out to be false, and certain symplectic groups are counterexamples. However, for `most' simple groups, the conjecture is true, so the question then is to determine exactly which simple groups have this property. In this paper, the authors sketch a proof that subgroups of odd order are pronormal in $\mathrm{PSp}_{2n}(q)$ for $q\equiv \pm 3 \mod 8$ and $n$ of the form $2^m(2^{2k}+1)$, where $m\ge 0$ and $k\ge 1$.
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 7002182
zbMATH Keywords: paranormal subgroups
zbMATH Keywords: Sylow \(p\)-subgroups
reviewed by: Robert A. Wilson
MaRDI profile type: Publication
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full work available at URL: s1064562418060029
OpenAlex ID: W2900245940
description: scientific article; zbMATH DE number 7002182