Injective endomorphism-quotient-balanced subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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Definition

A subgroup $H$ of a group $G$ is termed injective endomorphism-quotient-balanced if $H$ is a normal subgroup of $G$ and every injective endomorphism of $G$ sends $H$ to itself and induces an injective endomorphism on the quotient group $G/H$.

Relation with other properties

Stronger properties

• Finite injective endomorphism-invariant subgroup
  • Finite fully invariant subgroup
  • Characteristic subgroup of finite group
• Injective endomorphism-invariant subgroup with injective endomorphism-invariant complement

Weaker properties

• Injective endomorphism-invariant subgroup
• Characteristic subgroup
• Normal subgroup

Metaproperties

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

Further information: Quotient-balanced implies quotient-transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties