Some isomorphism invariants of integral group rings
type: Item
instance of: scholarly article
title: Some isomorphism invariants of integral group rings
zbMATH Open document ID: 0588.20003
DOI: 10.1216/RMJ-1985-15-2-451
author: Steve Kmet
author: Sudarshan K. Sehgal
published in: Rocky Mountain Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: This paper is concerned with the isomorphism problem for integral group rings: If the integral group rings \({\mathbb{Z}}(G)\) and \({\mathbb{Z}}(H)\) of the given groups G and H respectively are isomorphic, does this imply that G and H are isomorphic? For the group G let \(\{\gamma_ i(G)\}\) be its lower central series, \(\{\delta_ i(A)\}\) be its derived series, and \(D_ i(G)\) be its ith dimension subgroup over \({\mathbb{Z}}\). Assume that the torsion elements of G form a subgroup T(G) and the same for H. Define inductively: \(T_ 1(G)=T(G)\) and for \(i\geq 1\) \(T_{i+1}(G)=[G,T_ i(G)]\), the group generated by all commutators \(g^{-1}t^{-1}gt\), \(g\in G\), \(t\in T_ i(G).\) The main theorem of this paper gives, if \({\mathbb{Z}}(G)\simeq {\mathbb{Z}}(H)\), isomorphism statements of the quotients, \(T_ i(G)/T_{i+j}(G)\), \(D_ i(G)\cap T(G)/D_{i+j}(G)\cap T(G)\) for \(1\leq j\leq i+2\), \(\gamma_ i(T(G))/\gamma_{i+j}(T(G))\) for \(1\leq j\leq i\), \(\delta_ i(T(G))/\delta_{i+1}(T(G)),\) \(\delta_ i(T(G))/[G,\delta_ i(T(G))/[G,\delta_ i(T(G))]'\) for all i, with the corresponding ones of H. As a special case if G and H are torsion groups, then the theorem includes or extends in dimension (the i and j) results of \textit{T. Furukawa} [Math. J. Okayama Univ. 23, 125-130 (1981; Zbl 0473.20005)]. Furthermore, if \(D_ 6(G)=1\), then G'\(\simeq H'\). This improves a result by \textit{J. Ritter} and \textit{S. K. Sehgal} [Arch. Math. 40, 32-39 (1983; Zbl 0504.20003)] where it was assumed that G' is of exponent p.
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 3944119
zbMATH Keywords: isomorphism problem
zbMATH Keywords: integral group rings
zbMATH Keywords: lower central series
zbMATH Keywords: derived series
zbMATH Keywords: dimension subgroup
zbMATH Keywords: torsion groups
reviewed by: T. Akasaki
MaRDI profile type: Publication
full work available at URL: rmj-1985-15-2-451
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description: scientific article; zbMATH DE number 3944119