Odd order Hall subgroups of \(GL(n,q)\) and \(Sp(2n,q)\)
type: Item
instance of: scholarly article
title: Odd order Hall subgroups of \(GL(n,q)\) and \(Sp(2n,q)\)
zbMATH Open document ID: 0548.20013
DOI: 10.1007/BF01161703
author: Fletcher Gross
published in: Mathematische Zeitschrift
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173482
review text: Consider the group \(GL(n,q)\), \(q=p^ f\), \(p\) a prime. Let \(\omega\) be a set of primes with \(2,p\not\in\omega\). The author shows that if \(GL(n,q)\) possesses \(\omega\)-Hall subgroups, then any \(\omega\)-subgroup of \(GL(n,q)\) has a normal Abelian \(\tau\)-Hall subgroup where \(\tau=\omega-\{r\}\) and \(r\) is the smallest prime of \(\omega\). This in turn is used to prove that all \(\omega\)-Hall subgroups are conjugate. Moreover the author shows that a stronger conjecture on Hall subgroups is not true: For any set \(\omega\) of odd primes there is an \(n\) and a \(q\) such that the pair \(GL(n,q),\omega\) satisfies the above conditions, but there are \(\omega\)-subgroups which are not even isomorphic to any subgroup of an \(\omega\)-Hall subgroup of \(GL(n,q)\). Similar results are proved for \(Sp(2n,q)\). By this paper some results of \textit{E. L. Spitznagel} [Math. Z. 97, 259-290 (1967; Zbl 0183.03104)] are generalized.
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 3873580
zbMATH Keywords: Hall subgroups
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3873580