On Cohomologies and Representations of Groups with Normal Engel Subgroups

Abstract

Let \(\lambda, U\) be representations of a group \(G\) with a normal Engel subgroup \(N\). The paper studies triviality conditions for the cohomology group \(\mathcal{H}^1(G, \lambda, U)\) when \(\lambda\) and \(U\) are sectionally spectrally disjoint and examines some decompositions of the extension \(e(\lambda, U, \xi)\) of \(\lambda\) by \(U\) associated with non-trivial \((\lambda, U)\)-cocycles \(\xi\).