Recently, we have introduced a group invariant, which is related to the number of elements x and y of a nite group G such that x ^ y = 1 in the exterior square G ^ G of G. This number gives restrictions on the Schur multiplier of G and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form {h^m} ^ k of H ^ K such that {h^m} ^ k = 1, where m > 1 and H and K are arbitrary subgroups of G.
