On a construction related to the non-abelian tensor square of a group

LetG andG be isomorphic groups. We introduce and study a quotient (G) of the free productG *G which is a group extention of the non-abelian tensor squareG G. This seems to bring computational advantages to calculate this last group. Looking over as an operator in the class of groups we prove that it preserves properties of the argumentG such as finiteness, set of prime divisors, nilpotency and solvability. For a finitep-groupG we find a good polynomial bound for the order of (G).