Journal of the
Korean Mathematical Society JKMS

Recently, we have introduced a group invariant, which is 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 $m\ge1$ and $H$ and $K$ are arbitrary subgroups of $G$.

Keywords: $m$-th relative exterior degree, commutativity degree, exterior product, Schur multiplier, homological algebra

MSC numbers: Primary 20J99, 20D15; Secondary 20D60, 20C25