Commuting powers and exterior degree of finite groups

TL;DR

This paper generalizes the concept of exterior degree in finite groups by examining the number of elements of the form $h^m \,\wedge\\, k$ that are trivial in the exterior square, providing new insights into group invariants.

Contribution

It extends previous work on exterior degree by analyzing elements $h^m \,\wedge\\, k$ in the exterior square of subgroups, broadening the understanding of group invariants.

Findings

Generalized exterior degree to include elements $h^m \,\wedge\\, k$

Provided bounds and restrictions related to the Schur multiplier

Described classes of groups based on new exterior degree measures

Abstract

In [P. Niroomand, R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335-343] it is introduced a group invariant, related to the number of elements $x$ and $y$ of a finite group $G$, such that $x \wedge y = 1_{G \wedge G}$ in the exterior square $G \wedge G$ 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 $h^m \wedge k$ of $H \wedge K$ such that $h^m \wedge k = 1_{H \wedge K}$, where $m \ge 1$ and $H$ and $K$ are arbitrary subgroups of $G$.