The Nonabelian Tensor Squares and Homological Functors of Some 2-Engel Groups

Abstract

Originated in homotopy theory, the nonabelian tensor square is a special case of the nonabelian tensor product. The nonabelian tensor square of a group G, denoted as G7G is generated by the symbols g7h, for all g, hdG subject to the relations gg'7h = (gg'7gh) (g7h) and g7hh' = (g7h)(hg7hh'), for all g, g', h, h'dG where the action is taken to be conjugation. The homological functors of a group including J(G), d(G), the exterior square, the Schur multiplier, d(G), the symmetric square and J (G) d(G) are closely related to the nonabelian tensor square of the group. In this paper, the nonabelian tensor squares and homological functors of some 2-Engel groups will be presented.