An improved 3-local characterization of McL and its automorphism group.
type: Item
instance of: scholarly article
title: An improved 3-local characterization of McL and its automorphism group.
zbMATH Open document ID: 1345.20021
author: Gernot Stroth
author: Chris Parker
published in: Journal of Algebra
publication date: 2014-08-28T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.1077
reviewed by: Ali Reza Moghaddamfar
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6335122
zbMATH Keywords: finite simple groups
zbMATH Keywords: McLaughlin simple group
zbMATH Keywords: 3-local characterizations
zbMATH Keywords: local characteristic \(p\)
describes a project that uses: Magma
MaRDI profile type: Publication
OpenAlex ID: W2051270933
arXiv ID: 1201.1077
cites work: Q3708174
cites work: The Magma algebra system. I: The user language
cites work: Finite Groups Admitting Automorphisms with Prescribed Fixed Points
cites work: Q3684278
cites work: Q3996909
cites work: Finite Groups which Contain a Selfcentralizing Subgroup of Order 3
cites work: Q5577154
cites work: Q4426043
cites work: A 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups.
cites work: Strongly \(p\)-embedded subgroups.
cites work: A family of fusion systems related to the groups \(\mathrm{Sp}_4(p^a)\) and \(\mathrm G_2(p^a)\).
cites work: On finite groups with a certain Sylow normalizer. I
DOI: 10.1016/J.JALGEBRA.2014.02.011
Recommended article: A 3-local characterization of \(Co_3\).
Recommended article: A 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups.
Recommended article: Fusion systems on a Sylow 3-subgroup of the McLaughlin group
Recommended article: Some amalgams in characteristic \(3\) related to \(Co_ 1\)
Recommended article: A 3-local identification of two almost simple extensions of \(P\Omega_8^+(2)\).
Recommended article: A 2-local characterization of \(M_{12}\)
Recommended article: The 5-modular characters of the McLaughlin group and its covering group
Recommended article: A characterization of automorphism groups of simple \(K_3\)-groups.
Recommended article: A characterization of 3-local subgroups of Ly
Recommended article: Errata corrige: Some sporadic groups as Galois groups. II
review text: The authors give a \(3\)-local characterization of the McLaughlin sporadic simple group McL and its automorphism group \(\Aut(\mathrm{McL})\). As a matter of fact, this result extends earlier work of \textit{C. Parker} and \textit{P. Rowley} [J. Algebra 319, No. 4, 1752-1775 (2008; Zbl 1140.20015)] in which McL and \(\Aut(\mathrm{McL})\) are characterized by certain \(3\)-local information.NEWLINENEWLINE Main Theorem of this article (Theorem 1.1): Suppose that \(G\) is a finite group, \(S\in\mathrm{Syl}_3(G)\), \(Z=Z(S)\) and \(J\) is an elementary abelian subgroup of \(S\) of order \(3^4\). Further assume that (i) \(O^{3'}(N_G(Z))\approx 3_+^{1+4}.2\cdot\mathrm{Alt}(5)\); (ii) \(O^{3'}(N_G(J))\approx 3^4.\mathrm{Alt}(6)\); and (iii) \(C_G(O_3(C_G(Z)))\leqslant O_3(C_G(Z))\). Then \(G\cong\mathrm{McL}\) or \(\Aut(\mathrm{McL})\).NEWLINENEWLINE The difference between this theorem and the characterization theorem of Parker and Rowley is as follows. In fact, the main theorem of the former article essentially assumes that the group under investigation is of local characteristic 3, but the main theorem here only assumes that the group \(G\) has parabolic characteristic \(3\).
description: scientific article; zbMATH DE number 6335122