Homotopy theory and generalized dimension subgroups

Abstract

Let G be a group and R,S,T its normal subgroups. There is a natural extension of the concept of commutator subgroup for the case of three subgroups ‖R,S,T‖ as well as the natural extension of the symmetric product ‖r,s,t‖ for corresponding ideals r,s,t in the integral group ring Z[G]. In this paper, it is shown that the generalized dimension subgroup G∩(1+‖r,s,t‖) has exponent 2 modulo ‖R,S,T‖. The proof essentially uses homotopy theory. The considered generalized dimension quotient of exponent 2 is identified with a subgroup of the kernel of the Hurewicz homomorphism for the loop space over a homotopy colimit of classifying spaces.