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.