Groups with non-central dimension quotients
Let \(G\) be a group and let \(\Delta(G)\) be the augmentation ideal of the integral group ring \(\mathbb{Z} G\). The dimension subgroups \(D_n (G)\), \(n \geq 1\), are defined by \(D_n(G) = G \cap (1 + \Delta^n)\). While it is easy to see that \(D_n(G) \supseteq \gamma_n(G)\), for all \(n \geq 1\), where \(\gamma_n(G)\) is the \(n\)th term in the lower central series of \(G\), the identification of the quotients \(D_n(G)/\gamma_n(G)\) remains a challenging problem in the theory of groups. Whereas this quotient is always trivial for \(n \leq 3\), it is, in general, non-trivial for every \(n \geq 4\) [\textit{E. Rips}, Isr. J. Math. 12, 342-346 (1972; Zbl 0267.20018); \textit{N. Gupta}, Bull. Lond. Math. Soc. 22, No. 5, 453-456 (1990; Zbl 0721.20001)]. For an account of the work on dimension subgroups over the years see \textit{N. Gupta}, Free Group Rings [Contemp. Math. 66 (1987; Zbl 0641.20022)] and \textit{N. Gupta} [Groups, St. Andrews 1989, Lond. Math. Soc. Lect. Note Ser. 159, 214-249 (1991; Zbl 0742.20011)]. It is noted in Free Group Rings [loc. cit.], p. 85, Corollary 3.7, that for a metabelian group \(G\), \(D_{n-1}(G)/\gamma_n(G)\) (so, in particular, \(D_n(G)/\gamma_n(G))\) is central in \(G/\gamma_n(G)\). In the present paper the authors prove that \(D_n(G)/\gamma_n(G)\) is, in general, not central in \(G/\gamma_n(G)\). Groups with such a behaviour are shown to result from groups with non-trivial \(D_n(G)/\gamma_n(G)\) and an embedding theorem (Theorem 5) which states that, given a finitely generated nilpotent group \(H\) of class \(c\), an integer \(s \geq 1\) and \(1 \neq h_0 \in H\), for some \(m\) (depending on \(h_0\) and \(s\)) and any \(n \geq m\) there exists a nilpotent group \(G\) of class at most \(nc\) and an embedding \(\mu : H \to G\) such that \(\mu(H) \subseteq \gamma_{n- m + 1}(G)\) whereas \(\mu(h_0) \not\in Z_s(G)\), the \(s\)th upper central subgroup of \(G\).
scientific article; zbMATH DE number 825334