On the nilpotency class and solvability length of nonabelian tensor products of groups

Abstract.

The nonabelian tensor product $G \otimes H$ is defined for a pair of groups G and H, provided G and H act on each other in a compatible fashion. The goal of this paper is to give bounds on the nilpotency class and solvability length of $G \otimes H$, provided such information is given in context with G and H. The bounds are given in terms of D _H (G), the derivative subgroup of G afforded by the action of H on G, and D _G (H), the analogous subgroup of H. These derivative subgroups reduce to the commutator subgroup of G if G = H with all actions conjugation. We conclude with an example showing that, in contrast to the cases of the tensor square, the tensor product of a pair of groups of nilpotency class two is not necessarily abelian.