Symmetric Elements of Nonlinear Involutions in Group Rings

Abstract

Given an involution φ : G → G in a group G and a ring R, we study the extensions, not necessarily linear, to an involution ψ : RG → RG in the group ring RG. We investigate the symmetric elements, those α ∈ RG for which ψ(α) = α, and give necessary and sufficient conditions for the set of symmetric elements, (RG)ψ, to be a subring of RG. This work is a generalization of [6] and references therein where only linear extensions of the group involution are considered.

Group rings whose set of symmetric elements is Lie metabelian

“…linear extension of involutions of G, the commutativity of RG + was studied in [GDR12] and that of RG − in [Rapar].…”