The unit group of \(FS_4\).
type: Item
instance of: scholarly article
title: The unit group of \(FS_4\).
zbMATH Open document ID: 1156.16024
author: Manju Khan
author: J. B. Srivastava
author: Rajendra Kumar Sharma
published in: Acta Mathematica Hungarica
publication date: 2008-10-21T00:00:00Z
review text: Let \(FS_m\) denote a group algebra of the symmetric group \(S_m\) of degree \(m\) over a finite field \(F\). In [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 23, No. 2, 129-142 (2007; Zbl 1135.16034)] the authors characterized the unit group \(U(FS_3)\) of \(FS_3\). This work is continued here, they consider the group \(S_4\) and prove that if \(F\) is the field of \(p^k\) elements, where \(p>3\), then the unit group \(U(FS_4)\) is isomorphic to \[ \text{GL}(3,F)\times\text{GL}(3,F)\times\text{GL}(2,F)\times U(F)\times U(F), \] where \(\text{GL}(n,F)\) denotes the general linear group of degree \(n\) over \(F\). Furthermore, if \(p\leq 3\), then \(U(FS_4)\) is an extension of the group \(G\) by the group \(1+J(FS_4)\), where \(G\) is either \(\text{GL}(2,F)\times U(F)\) or \(\text{GL}(3,F)\times\text{GL}(3,F)\times U(F)\times U(F)\) depending on whether \(p\) is equal to \(2\) or \(3\), and \(J(FS_4)\) is the Jacobson radical of \(FS_4\). Also the structure of \(1+J(FS_4)\) is determined in both cases, and the characterization is hereby complete.
reviewed by: Tibor Juhász
Mathematics Subject Classification ID: 16U60
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C30
zbMATH DE Number: 5355075
zbMATH Keywords: symmetric groups
zbMATH Keywords: unit groups
zbMATH Keywords: group algebras
MaRDI profile type: Publication
full work available at URL: s10474-007-6169-4
OpenAlex ID: W2022312318
cites work: A characterization of units in zs4
cites work: Q4145843
cites work: On Congruence Subgroups and Units in ZS4
cites work: Q5426267
DOI: 10.1007/S10474-007-6169-4
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description: scientific article; zbMATH DE number 5355075