Identifications in modular group algebras.
type: Item
instance of: scholarly article
title: Identifications in modular group algebras.
zbMATH Open document ID: 1148.20002
author: Hamid Usefi
published in: Journal of Pure and Applied Algebra
publication date: 2008-07-10T00:00:00Z
review text: Let \(FG\) be the group algebra of a finite \(p\)-group over a field of prime characteristic \(p\). The author provides a new proof of the theorem on the so called Jennings basis for the factors of the powers of the augmentation ideal \(\omega\) in terms of the dimension subgroups \(D_n\), and of the theorem of Quillen which states that the graded algebra associated to the group algebra \(FG\) and the enveloping algebra of the restricted Lie algebra associated to the dimension subgroups are isomorphic algebras. The main results of the paper are as follows: for a subgroup \(S\) of \(G\) \[ G\cap(1+\omega(G)\omega^n(S))=D_{n+1}(S)\quad\text{and}\quad G\cap(1+\omega^2(G)\omega^n(S))=D_{n+2}(S)D_{n+1}(S\cap D_2(FG)). \] The analogue of the second statement for integral group rings of free groups has been obtained recently by \textit{T. Hurley} and \textit{S. Sehgal} [in Commun. Algebra 28, No. 2, 1051-1059 (2000; Zbl 0963.20002)].
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20F40
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 5299436
zbMATH Keywords: group algebras
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: augmentation ideals
zbMATH Keywords: dimension subgroups
zbMATH Keywords: Jennings bases
zbMATH Keywords: enveloping algebras
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.01.006
OpenAlex ID: W2056599856
cites work: Q3999421
cites work: On the Group Ring
cites work: Free differential calculus. I: Derivation in the free group ring. II: The isomerphism problem of groups. III: Subgroups
cites work: Cohomological topics in group theory.
cites work: Q3934684
cites work: Identifications in a free group
cites work: Groups related to fox subgroups
cites work: The Group Ring Of a Class Of Infinite Nilpotent Groups
cites work: Q3755641
cites work: Combinatorial methods. Free groups, polynomials, and free algebras
cites work: Group rings and their augmentation ideals
cites work: On the associated graded ring of a group ring
cites work: Restricted Lie Algebras and Their Envelopes
cites work: Q4189444
cites work: Q3741786
DOI: 10.1016/J.JPAA.2008.01.006
description: scientific article; zbMATH DE number 5299436