On Baer Invariants of Pairs of Groups

TL;DR

This paper extends the concept of Baer invariants to pairs of groups using simplicial group theory, exploring their behavior under free products and direct limits, and proving specific properties for nilpotent multipliers.

Contribution

It introduces a generalized Baer invariant for pairs of groups relative to any variety, expanding the theoretical framework and analyzing its properties under various group operations.

Findings

Baer invariants of pairs of groups are developed using simplicial groups.

Behavior of Baer invariants under free product and direct limit is analyzed.

Nilpotent multiplier commutes with free product of finite coprime order groups.

Abstract

In this paper, we use the theory of simplicial groups to develop the Schur multiplier of a pair of groups $(G,N)$ to the Baer invariant of it, $\mathcal{V}M(G,N)$, with respect to an arbitrary variety $\mathcal{V}$. Moreover, we present among other things some behaviors of Baer invariants of a pair of groups with respect to the free product and the direct limit. Finally we prove that the nilpotent multiplier of a pair of groups does commute with the free product of finite groups of mutually coprime orders.