A solution of the dimension subgroup problem
Let \({\mathbb{Z}}G\) be the integral group ring of a group G. For \(n\geq 1\) the nth dimension subgroup \(D_ n(G)\) is defined as \(D_ n(G)=G\cap (1+\Delta^ n(G))\) where \(\Delta\) (G) is the augmentation ideal, evidently \(D_ n(\gamma)\geq \gamma_ n(G)\) where \(\gamma_ n(G)\) is the nth term of the lower central series. Since it is now known that, in general, \(D_ n(G)\neq \gamma_ n(G)\) (n\(\geq 4)\), the determination of the factor groups \(D_ n(G)/\gamma_ n(G)\) has become the so-called dimension subgroup problem. For metabelian p-groups the author had extended an earlier result of \textit{J. A. Sjogren} [J. Pure Appl. Algebra 14, 175-194 (1979; Zbl 0395.20019)] to show that \(D_ n(G)=\gamma_ n(G)\) for all \(n\leq 2p-1\) [ibid. 51, 241-249 (1988; Zbl 0649.20004)]. In this paper he establishes that if G is a finite metabelian group then \(D_ n(G)/\gamma_ n(G)\) has exponent which is a power of 2, and that if G is a finitely generated metabelian p-group, p odd, then \(D_ n(G)=\gamma_ n(G)\) for all \(n\geq 1\).
scientific article; zbMATH DE number 4204638