Homomorphism extension problem for subdirect products of finite groups

TL;DR

This paper investigates the conditions under which homomorphisms from subdirect products of finite groups to abelian groups can be extended, focusing on cases with trivial Schur multipliers and twisted diagonal subgroups.

Contribution

It provides new criteria for homomorphism extensibility in subdirect products of finite groups, especially involving Goursat quotients and twisted diagonals.

Findings

Homomorphisms with trivial Schur multipliers are extensible.

Extension properties are analyzed for subdirect products with twisted diagonal subgroups.

The functoriality of extensibility is explored.

Abstract

Motivated by the simplification of decomposition formulas for fibred bisets, we study the homomorphism extension problem for subdirect products of finite groups when the codomain is an abelian group satisfying certain hypothesis. We prove that every homomorphism of subdirect products whose Goursat quotients have trivial Schur multipliers is extensible. We also examine the case where subdirect products contain a twisted diagonal subgroup and investigate the functoriality of the extensibility property.