Some isomorphism invariants of integral group rings
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.
scientific article; zbMATH DE number 3944119
scientific article