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Homomorphism extension problem for subdirect products of finite groups

İsmail Alperen Öğüt

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.

Homomorphism extension problem for subdirect products of finite groups

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.
Paper Structure (5 sections, 15 theorems, 25 equations)

This paper contains 5 sections, 15 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Necessary and Sufficient Condition for Extensibility
  4. Extensibility via criteria on $q(U)$
  5. The case of containment of twisted diagonals

Key Result

Theorem 1.2

Let $p$ be a prime, $G,H$ finite groups, $U\leq G\times H$ a subdirect product. Then $U$ is $p$-extensible if one of the following conditions hold:

Theorems & Definitions (21)

  • Theorem 1.2
  • Lemma 2.1: Goursat
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 11 more