The modular version of Maschke's theorem for normal abelian \(p\)-Sylows
type: Item
instance of: scholarly article
title: The modular version of Maschke's theorem for normal abelian \(p\)-Sylows
zbMATH Open document ID: 0849.16031
DOI: 10.1016/0022-4049(95)00047-X
author: Murray Gerstenhaber
author: Mary Schaps
published in: Journal of Pure and Applied Algebra
publication date: 1996-11-04T00:00:00Z
review text: The authors prove the Donald-Flanigan conjecture in characteristic \(p\) for groups having a normal abelian Sylow \(p\)-subgroup. Let \(p\) be a prime number, let \(G\) be a finite group and let \(k\) be an algebraically closed field of characteristic \(p\). Then, Donald and Flanigan conjectured that there is a deformation of the group algebra \(kG\) to a separable algebra for which the dimensions of the matrix algebras which occur are the same as for the group algebra over the complex numbers. Donald and Flanigan proved this in the abelian case and the second author of the paper under review proved the conjecture for groups with cyclic Sylow \(p\)-subgroups. Let \(P\) be a finite group and let \(H\) be a group of automorphisms of \(P\) such that \(p\) does not divide the order of \(H\). Then, the authors show that an \(H\)-linear separable deformation of \(kP\) can be extended to the semidirect product \(k(P\rtimes H)\). Let \(G\) be a finite group with abelian normal Sylow \(p\)-subgroup \(P\), then \(G\simeq P\rtimes H\) for a \(p'\)-subgroup \(H\) of \(G\). Then, the authors construct explicitly an \(H\)-linear separable deformation of \(kP\), hence confirming Donald-Flanigan's conjecture for \(G\).
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 897491
zbMATH Keywords: Donald-Flanigan conjecture
zbMATH Keywords: normal Abelian Sylow \(p\)-subgroup
zbMATH Keywords: finite groups
zbMATH Keywords: group algebras
zbMATH Keywords: separable algebras
zbMATH Keywords: matrix algebras
zbMATH Keywords: group of automorphisms
zbMATH Keywords: separable deformations
zbMATH Keywords: semidirect products
MaRDI profile type: Publication
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full work available at URL: 0022-4049(95)00047-x
OpenAlex ID: W2065851085
description: scientific article; zbMATH DE number 897491