An identification of \(Co_1\).
type: Item
instance of: scholarly article
title: An identification of \(Co_1\).
zbMATH Open document ID: 1161.20011
author: M. Reza Salarian
published in: Journal of Algebra
publication date: 2008-10-17T00:00:00Z
review text: The main result of the paper under review is the following Theorem (Theorem 1.1 of paper): Theorem 1.1. Let \(G\) be a finite group and \(S\in\text{Syl}_3(G)\). Let \(H_1\) and \(H_2\) be two subgroups of \(G\) containing \(S\) such that (i) \(H_1=N_G(Z(O_3(H_1)))\), \(O_3(H_1)\) is an extraspecial group of order \(3^5\), \(H_1 /O_3(H_1)\cong\text{SP}_4(3):2\) and \(C_{H_1}(O_3(H_1))=Z(O_3(H_1))=\langle t\rangle\). (ii) \(O_3(H_2)\) is an elementary Abelian group of order \(3^6\) and \(H_2/O_3(H_2)\cong 2M_{12}\). (iii) \((H_1\cap H_2)/O_3(H_2)\) is an extension of an elementary Abelian group of order \(9\) by \(\text{GL}_2(3)\times Z_2\). -- Then \(G\) is isomorphic to \(Co_1\). For the proof of this Theorem, the author determines the structure of the centralizer of a \(2\)-central involution in the group \(G\). He finds an involution \(z\) in \(G\) such that \(C_G(z)\) is an extension of an extraspecial \(2\)-group of order \(2^9\) by \(\Omega_8^+(2)\) and the main results follow by applying Lemma 49.15 of the book of \textit{M. Aschbacher} [Sporadic groups. Cambridge Tracts in Mathematics 104. Cambridge: Cambridge University Press (1994; Zbl 0804.20011)].
reviewed by: A. Iranmanesh
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5353581
zbMATH Keywords: finite sporadic simple groups
zbMATH Keywords: 2-central involutions
zbMATH Keywords: extraspecial 2-groups
zbMATH Keywords: 3-local subgroups
zbMATH Keywords: Conway group \(Co_1\)
zbMATH Keywords: centralizers of involutions
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.05.006
OpenAlex ID: W2464337238
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DOI: 10.1016/J.JALGEBRA.2008.05.006
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description: scientific article; zbMATH DE number 5353581