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 ) ≅ H / Z ∧ ⁢ ( H ) G/Z^{\wedge}(G)\cong H/Z^{\wedge}(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.