Invariance of the Schur multiplier, the Bogomolov multiplier and the minimal number of generators under a variant of isoclinism

We introduce the q-Bogomolov multiplier as a generalization of the Bogomolov multiplier, and we prove that it is invariant under q-isoclinism. We prove that the q-Schur multiplier is invariant under q-exterior isoclinism, and as an easy consequence, we prove that the Schur multiplier is invariant under exterior isoclinism. We also prove that if G and H are p-groups with G=Z^{^}.G/ Š H=Z^{^}.H/, then the cardinalities of the minimal number of generators of G and H are the same. Moreover, we prove some structural results about non-abelian q-tensor square of groups.