Identifications in modular group algebras.
Let \(FG\) be the group algebra of a finite \(p\)-group over a field of prime characteristic \(p\). The author provides a new proof of the theorem on the so called Jennings basis for the factors of the powers of the augmentation ideal \(\omega\) in terms of the dimension subgroups \(D_n\), and of the theorem of Quillen which states that the graded algebra associated to the group algebra \(FG\) and the enveloping algebra of the restricted Lie algebra associated to the dimension subgroups are isomorphic algebras. The main results of the paper are as follows: for a subgroup \(S\) of \(G\) \[ G\cap(1+\omega(G)\omega^n(S))=D_{n+1}(S)\quad\text{and}\quad G\cap(1+\omega^2(G)\omega^n(S))=D_{n+2}(S)D_{n+1}(S\cap D_2(FG)). \] The analogue of the second statement for integral group rings of free groups has been obtained recently by \textit{T. Hurley} and \textit{S. Sehgal} [in Commun. Algebra 28, No. 2, 1051-1059 (2000; Zbl 0963.20002)].
scientific article; zbMATH DE number 5299436
scientific article