Characterizing finite quasisimple groups by their complex group algebras.
type: Item
instance of: scholarly article
title: Characterizing finite quasisimple groups by their complex group algebras.
zbMATH Open document ID: 1303.20001
author: Hung Ngoc Nguyen
author: H. P. Tong-Viet
published in: Algebras and Representation Theory
publication date: 2014-06-23T00:00:00Z
review text: A fundamental question in the representation theory of finite groups is whether one can recover a group or some of its properties from its complex group algebra. It is believed that nonabelian simple groups or more generally quasisimple groups have a strong connection with their complex group algebras. A finite group \(L\) is said to be quasisimple if \(L\) is perfect and \(L/Z(L)\) is nonabelian simple, in which case we also say that \(L\) is a cover of \(L/Z(L)\). It has been proved recently [\textit{H. N. Nguyen}, Isr. J. Math. 195, Part B, 973-998 (2013; Zbl 1294.20004)] that a quasisimple classical group \(L\) is uniquely determined up to isomorphism by the structure of \(\mathbb CL\), the complex group algebra of \(L\), when \(L/Z(L)\) is not isomorphic to \(\text{PSL}_3(4)\) or \(\text{PSU}_4(3)\). In this paper, the authors establish a similar result for these two open cases and also for covers with nontrivial center of simple groups of exceptional Lie type and sporadic groups. Together with the main results of \textit{H. P. Tong-Viet} [Monatsh. Math. 166, No. 3-4, 559-577 (2012; Zbl 1255.20006); Algebr. Represent. Theory 15, No. 2, 379-389 (2012; Zbl 1252.20005)], they obtain that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of its complex group algebra.
reviewed by: Nan Ji-Zhu
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20C34
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 6306467
zbMATH Keywords: complex group algebras
zbMATH Keywords: finite quasisimple groups
zbMATH Keywords: finite exceptional groups
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: sporadic simple groups
zbMATH Keywords: quasisimple classical groups
zbMATH Keywords: finite simple groups
zbMATH Keywords: isomorphism problem
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s10468-012-9400-0
OpenAlex ID: W2091843462
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DOI: 10.1007/S10468-012-9400-0
description: scientific article; zbMATH DE number 6306467