The Schur multiplier of central product of groups

TL;DR

This paper investigates the second cohomology group, specifically the Schur multiplier, of a central product of groups, relating it to the cohomology of related quotient groups, and refines existing results for complex coefficients.

Contribution

It provides new insights into the Schur multiplier of central products, extending previous results and offering a more detailed understanding of their cohomological structure.

Findings

Relates second cohomology of central product to quotients of component groups.

Refines known results for the case when coefficients are complex numbers.

Provides a framework for understanding Schur multipliers in central group constructions.

Abstract

Let $G$ be a central product of two groups $H$ and $K$. We study second cohomology group of $G$, having coefficients in a divisible abelian group $D$ with trivial $G$-action, in terms of the second cohomology groups of certain quotients of $H$ and $K$. In particular, for $D = \mathbb{C}^{*}$, some of our results provide a refinement of results from [Some groups with non-trivial multiplicators, Math. Z. {\bf 120 } (1971), 307-308] and [On the Schur multiplicator of a central quotient of a direct product of groups, J. Pure Appl. Algebra {\bf 3} (1973), 73-82].