# Finite injective endomorphism-invariant subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: injective endomorphism-invariant subgroup with a group property (itself viewed as a subgroup property): finite group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **finite injective endomorphism-invariant subgroup** if it satisfies the following equivalent conditions:

- It is finite as a group and is an injective endomorphism-invariant subgroup of the whole group: every injective endomorphism of the whole group sends the subgroup to itself.
- It is finite as a group and is an injective endomorphism-quotient-balanced subgroup of the whole group: every injective endomorphism of the whole group induces an injective endomorphism on the quotient group.