A note on polynomial maps
For a group G let ZG denote its integral group ring and \(\Delta(G)\) the augmentation ideal. Extending a result of \textit{G. M. Bergman} and \textit{W. Dicks} [J. Algebra 36, 193-211 (1975; Zbl 0271.16017)], the authors prove that if H and K are subgroups of a group G, then \[ G\cap (1+ZG\Delta(H)\Delta(K))=\gamma_ 2(H^ K\cap K) \] where \(H^ K\) is the subgroup of G generated by all elements \(k^{-1}hk\), \(h\in H\), \(k\in K\) and \(\gamma_ i(M)\) denotes the ith term of the lower central series of the group M. [See also Theorem 1.14 on page 6 of \textit{N. Gupta}, ``Free Group Rings'' (Contemp. Math. 66, 1987; Zbl 0641.20022).] The second main result of the paper states that if G is a group with normal subgroups H and K such that \(G=HK\) and \(H\cap K\) is contained in the centre \(\zeta(G)\), then \[ G\cap (1+\Delta^ 3(G)+\Delta (H)\Delta (G))=\gamma_ 2(H)\gamma_ 3(G). \] This latter result extends a result (Theorem 3.11) of \textit{R. Sandling} [J. Algebra 21, 216-231 (1972; Zbl 0233.20001)] which states that \[ G\cap (1+\Delta^ 3(G)+\Delta (\zeta (G))\Delta (G))=\gamma_ 3(G). \]
scientific article; zbMATH DE number 4063351