Another proof of Gluck's theorem
type: Item
instance of: scholarly article
title: Another proof of Gluck's theorem
zbMATH Open document ID: 1007.20002
author: Hiroshi Matsuyama
published in: Journal of Algebra
publication date: 2003-01-05T00:00:00Z
review text: Let \(G\) be a permutation group acting on a finite set \(\Omega\). A subset \(S\) of \(\Omega\) is called \(G\)-regular if the stabilizer \(\{g\in G\mid Sg=S\}\) is the identity subgroup. \textit{D. Gluck}'s theorem [Can. J. Math. 35, 59-67 (1983; Zbl 0509.20002)] states that \(G\) has a regular subset in \(\Omega\) if \(G\) is of odd order. In the present paper an alternative proof of Gluck's theorem is given which avoids \textit{T. R. Wolf}'s result on primitive solvable groups [Can. J. Math. 34, 1097-1111 (1982; Zbl 0493.20027)] but still relies on the odd order theorem.
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 1740522
zbMATH Keywords: finite permutation groups
zbMATH Keywords: regular subsets
zbMATH Keywords: solvable groups
zbMATH Keywords: groups of odd order
MaRDI profile type: Publication
full work available at URL: jabr.2001.9026
OpenAlex ID: W2087411723
cites work: Q4320507
cites work: Solvability of groups of odd order
cites work: Trivial Set-Stabilizers in Finite Permutation Groups
cites work: Solvable and Nilpotent Subgroups of GL(n,qm)
DOI: 10.1006/JABR.2001.9026
description: scientific article; zbMATH DE number 1740522