Characteristically complemented subgroup

Symbol-free definition
A subgroup of a group is termed a characteristically complemented subgroup or characteristic retract if it satisfies the following equivalent conditions:


 * 1) There is a defining ingredient::retraction (viz an idempotent endomorphism) on the group, whose image is that subgroup, and whose kernel is a defining ingredient::characteristic subgroup.
 * 2) There is a characteristic subgroup that is a permutable complement to it.
 * 3) There is a characteristic subgroup that is a lattice complement to it.

The corresponding characteristic subgroup (there may be more than one such) is termed a complemented characteristic subgroup.

Stronger properties

 * Weaker than::Characteristically complemented characteristic subgroup
 * Weaker than::Characteristically complemented normal subgroup

Weaker properties

 * Stronger than::Quasicharacteristic retract
 * Stronger than::Retract
 * Stronger than::Permutably complemented subgroup
 * Stronger than::Lattice-complemented subgroup

Metaproperties
If $$H \le K \le G$$ are groups such that $$H$$ is characteristically complemented in $$K$$ and $$K$$ is characteristically complemented in $$G$$, then $$H$$ is characteristically complemented in $$G$$. This follows essentially from the fact that characteristicity is quotient-transitive.