Solution of the twisting problem for skew group algebras
type: Item
instance of: scholarly article
title: Solution of the twisting problem for skew group algebras
zbMATH Open document ID: 0836.16016
DOI: 10.1007/BF02761659
author: Eli Aljadeff
author: Derek J. S. Robinson
published in: Israel Journal of Mathematics
publication date: 1996-04-16T00:00:00Z
review text: Let \(G\) be a finite group, \(K\) a field of prime characteristic \(p\), and \(\tau:G\to\text{Aut}(K)\) an action of \(G\) on \(K\) with kernel \(H\). If \(p\) divides the order of the subgroup \(H\), then the skew group ring \(KG\) is not semisimple. The authors [J. Pure Appl. Algebra 94, 1-15 (1994; Zbl 0809.16029)] proved that if there exists a class \(\alpha\in H^2(G,K^*)\) which ``twists'' the skew group ring \(K^\tau G\) into a semisimple crossed product \(K^\tau_\alpha G\), then \(H/O_{p'}(H)\) is an abelian group whose rank does not exceed the \(p\)-degree of \(K\) over \(K^p\). In this paper they prove that this condition is sufficient. Therefore, there is a semisimple crossed product of \(G\) over \(K\) if and only if \(G\) has a normal subgroup \(H\) such that \(G/H\) is isomorphic with a subgroup of \(\text{Aut}(K)\) and \(H/O_{p'}(H)\) is an abelian group with rank \(r\) not greater than the \(p\)-degree of \(K\) over \(K^p\).
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 820111
zbMATH Keywords: finite groups
zbMATH Keywords: actions
zbMATH Keywords: skew group rings
zbMATH Keywords: semisimple crossed products
zbMATH Keywords: normal subgroups
zbMATH Keywords: \(p\)-degree
reviewed by: S. V. Mihovski
MaRDI profile type: Publication
cites work: Semisimple algebras, Galois actions and group cohomology
cites work: Global dimensions of crossed products
cites work: Endliche Gruppen I
cites work: Q3992502
cites work: Q3995022
cites work: Q3856855
description: scientific article; zbMATH DE number 820111