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

TL;DR

This paper introduces the $q$-Bogomolov multiplier and demonstrates its invariance under $q$-isoclinism, along with invariance results for the $q$-Schur multiplier and minimal generator number in certain groups.

Contribution

It generalizes the Bogomolov multiplier to a $q$-version and proves its invariance under $q$-isoclinism, also establishing invariance of the $q$-Schur multiplier and minimal generator count.

Findings

The $q$-Bogomolov multiplier is invariant under $q$-isoclinism.

The $q$-Schur multiplier is invariant under $q$-exterior isoclinism.

Groups with isomorphic $Z^{igwedge}$ quotients have the same minimal number of generators.

Abstract

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 $G$, $H$ are $p$-groups and $G/Z^{\wedge}(G)\cong H/Z^{\wedge}(H)$, then the cardinality of the minimal number of generators of $G$ and $H$ are the same. Moreover we prove some structural results about $q$-nonabelian tensor square of groups.