The exterior degree of a pair of finite groups

TL;DR

This paper introduces the exterior degree of a pair of finite groups, exploring its relation to other group invariants like the Schur multiplier and relative commutative degree.

Contribution

It defines the exterior degree for pairs of groups and investigates its connections with existing concepts such as capability and the Schur multiplier.

Findings

Established relations between exterior degree and relative commutative degree

Linked exterior degree to the capability of group pairs

Connected exterior degree to the Schur multiplier

Abstract

The exterior degree of a pair of finite groups $(G,N)$, which is a generalization of the exterior degree of finite groups, is the probability for two elements $(g,n)$ in $(G,N)$ such that $g\wedge n=1$. In the present paper, we state some relations between this concept and the relative commutatively degree, capability and the Schur multiplier of a pair of groups.