Injective endomorphism-invariant subgroup

Symbol-free definition
A subgroup of a group is termed injective endomorphism-invariant if every injective endomorphism of the whole group takes the subgroup to within itself.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed injective endomorphism-invariant if for any injective endomorphism $$\sigma$$ of $$G$$, the image of $$H$$ under $$\sigma$$ is contained inside $$H$$.

Stronger properties

 * Weaker than::Fully characteristic subgroup
 * Weaker than::Isomorph-free subgroup
 * Weaker than::Isomorph-containing subgroup
 * Weaker than::Intermediately injective endomorphism-invariant subgroup

Weaker properties

 * Stronger than::Characteristic subgroup:
 * Stronger than::Normal subgroup

Related properties

 * Strictly characteristic subgroup: This is the invariance property with respect to surjective, rather than injective, endomorphisms.

Metaproperties
The property of being injective endomorphism-invariant is transitive on account of its being a.

The trivial subgroup is injective endomorphism-invariant because every endomorphism (injective or not) must take it to itself.

The whole group is also clearly injective endomorphism-invariant.

An arbitrary intersection of injective endomorphism-invariant subgroups is injective endomorphism-invariant. This follows on account of injective endomorphism-invariance being an.

An arbitrary join of injective endomorphism-invariant subgroups is injective endomorphism-invariant. This follows on account of injective endomorphism-invariance being an endo-invariance property.