On the structure of the augmentation quotients relative to an \(N_ p\)- series
For a group G with lower central series \(\{G_ i\}\), let \(W_ n(G)=\sum\otimes^{n}_{i=1}Sp^{a_ i}(G_ i/G_{i+1}),\) where \(Sp^{a_ i}(G_ i/G_{i+1})\) is the \(a_ i\)-th symmetric power of \(G_ i/G_{i+1}\), and the sum extends over sequences \(a_ 1,...,a_ n\) of nonnegative integers such that \(\sum ia_ i=n\). Let \(Q_ n(G)\) be the augmentation quotient \(I(G)^ n/I(G)^{n+1}\), where I(G) is the augmentation ideal of ZG. The author first shows that for a finite group of prime exponent p, \(Q_ n(G)\cong W_ n(G)/R_{n+1},\) for \(n\geq 1\). \(R_{n+1}\) is a certain subgroup of \(W_ n(G)\) that is described precisely. As corollaries, the author obtains a result of \textit{P. M. Cohn} [Proc. Lond. Math. Soc., III. Ser. 2, 297-310 (1952; Zbl 0047.025), with correction at the end of the volume], and the fact that \(Q_ n(G)\cong W_ n(G)\) for \(n<p\), provided that \(\{G_ i\}\) is an \(N_ p\)-series (i.e., \([G_ i,G_ j]\subseteq G_{i+j}\) for all i,\(j\geq 1\) and \(G^ p_ i\subseteq G_{ip}\) for all \(i\geq 1)\). Second, groups where \(\{G_ i\}\) is an \(N_ p\)-series are discussed further. For such groups, it is shown that \(Q_ p(G)\subseteq W_ p(G)\), \(Q_{p+1}(G)\) is a quotient of \(W_{p+1}(G)\) and \(D_ n(G)=G_ n\) for all \(n\geq 1\).
scientific article; zbMATH DE number 3873588
scientific article