TY - JOUR

T1 - Homotopy theory and generalized dimension subgroups

AU - Ivanov, Sergei O.

AU - Mikhailov, Roman

AU - Wu, Jie

PY - 2017/8/15

Y1 - 2017/8/15

N2 - 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.

AB - 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.

KW - Generalized dimension subgroups

KW - Group ring

KW - Homotopy groups

KW - Simplicial groups

UR - http://www.scopus.com/inward/record.url?scp=85019188646&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2017.04.012

DO - 10.1016/j.jalgebra.2017.04.012

M3 - Article

AN - SCOPUS:85019188646

VL - 484

SP - 224

EP - 246

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -