Finite injective endomorphism-invariant subgroup

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

Definition

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

1. 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.
2. 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.

Relation with other properties

Stronger properties

• Finite fully invariant subgroup
• Characteristic subgroup of finite group

Weaker properties

• Finite characteristic subgroup