# Complemented homomorph-containing subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: complemented normal subgroup and homomorph-containing subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **complemented homomorph-containing subgroup** if it satisfies the following equivalent conditions:

- It is both a complemented normal subgroup and a homomorph-containing subgroup of the whole group.
- It is both a complemented normal subgroup and a normal subgroup having no nontrivial homomorphism to its quotient group.
- It is both a permutably complemented subgroup and a homomorph-containing subgroup of the whole group.
- It is both a lattice-complemented subgroup and a homomorph-containing subgroup of the whole group.