A characterization of \(M_{22}\)
type: Item
instance of: scholarly article
title: A characterization of \(M_{22}\)
zbMATH Open document ID: 0882.20010
author: Chao Ku
published in: Journal of Algebra
publication date: 1998-03-09T00:00:00Z
review text: This paper, using modern techniques, presents a simplified local characterization of the Mathieu simple group \(M_{22}\). The main result is the following: Let \(G\) be a finite group with an involution \(z\) such that \(H=C_G(z)\) contains a normal extraspecial group \(Q\) of order 32 such that \(Z(Q)=\langle z\rangle\), \(H/Q\cong Z_2\times S_3\), \(C_H(Q)\leq Q\), \(z^G\cap Q\neq\{z\}\) and \(C_Q(X)=\langle z\rangle\) for any \(X\in\text{Syl}_3(H)\). Choosing \(g\in G-H\) such that \(z^g\in Q\), it is known that \(Q\cap Q^g\) is elementary abelian of order 4 or 8. If also \(Q\cap Q^g\) is of order 4, then say that \(G\) is of type \(M_{22}\). Main Theorem. If \(G\) is of type \(M_{22}\) then either \(G\cong M_{22}\) or \(G\) is isomorphic to the split extension of \(E_{16}\) by \(A_6\).
Mathematics Subject Classification ID: 20D08
zbMATH DE Number: 1014044
zbMATH Keywords: Mathieu group \(M_{22}\)
zbMATH Keywords: local characterization
zbMATH Keywords: finite groups
zbMATH Keywords: involutions
zbMATH Keywords: normal extraspecial subgroups
reviewed by: Morton E. Harris
MaRDI profile type: Publication
full work available at URL: jabr.1997.6792
OpenAlex ID: W2464892351
cites work: Q3708174
cites work: Q4295765
cites work: Q3128706
cites work: Central elements in core-free groups
cites work: A characterization of the Mathieu simple groups. I, II
DOI: 10.1006/JABR.1997.6792
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description: scientific article; zbMATH DE number 1014044