Strictly characteristic subgroup

Definition
A subgroup of a group is termed strictly characteristic or distingushed if it satisfies the following equivalent conditions:

Related properties

 * Injective endomorphism-invariant subgroup
 * Retraction-invariant subgroup, retraction-invariant normal subgroup, retraction-invariant characteristic subgroup

Formalisms
The property of being strictly characteristic is second-order. A subgroup $$H$$ is strictly characteristic in a group $$G$$ if:

$$\ \forall g \in H, \sigma \in G^G, (\ \forall \ a,b \in G, \sigma(ab) = \sigma(a)\sigma(b)) \land (\ \forall a \in G, \exists b: \sigma(b) = a) \implies \sigma(g) \in H$$

Note that the two conditions checked parenthetically are respectively the conditions of being an endomorphism and being surjective.

Origin of the concept
The concept has been explored under two names: strictly characteristic subgroup and distinguished subgroup. The first term has been used in Bourbaki's texts in a more general context of algebras.

The term strictly characteristic was used by Reinhold Baer in his paper The Higher Commutator Subgroups of a Group where he compares invariance properties like being normal, characteristic, strictly characteristic and fully characteristic.