Retraction-invariant subgroup

Symbol-free definition
A subgroup of a group is termed retraction-invariant if any retraction (viz, idempotent endomorphism) of the whole group takes the subgroup to within itself.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed retraction-invariant if, given any retraction (viz, idempotent endomorphism) $$\sigma$$ of $$G$$, $$\sigma(H) \le H$$.

A retraction is an idempotent endomorphism, viz $$\sigma$$ is a retraction if and only if $$\sigma(\sigma(g)) = \sigma(g)$$ for all $$g$$ in $$G$$.

Stronger properties

 * Weaker than::Fully invariant subgroup

Conjunction with other properties

 * Weaker than::Retraction-invariant retract is the conjunction with the property of being a retract.
 * Weaker than::Retraction-invariant direct factor is the conjunction with the property of being a direct factor.
 * Weaker than::Retraction-invariant central factor is the conjunction with the property of being a central factor.
 * Weaker than::Retraction-invariant normal subgroup is the conjunction with the property of being a normal subgroup.

Weaker properties

 * Stronger than::Direct projection-invariant subgroup
 * Stronger than::Retract-transfering subgroup: In other words, if $$H$$ is a retraction-invariant subgroup of $$G$$ and $$K$$ is a retract of $$G$$, then $$H \cap K$$ is a retract of $$H$$.

Incomparable with normality
Note that there are retraction-invariant subgroups which are not normal. In fact, in a simple group, every subgroup is retraction-invariant, although none except the trivial subgroup and the whole group are normal.

Further, we have examples of normal subgroups that are not retraction-invariant. For instance, the copy of $$G$$ in $$G \times G$$ is not invariant under the retraction $$(g,h) \mapsto (g,g)$$.

Metaproperties
Any retraction-invariant subgroup of a retraction-invariant subgroup is retraction-invariant. This easily follows on account of retraction-invariance being a balanced subgroup property, that is, from the fact that its restriction formal expression has the same thing on the left side and on the right side.

The whole group and the trivial subgroup are clearly retraction-invariant.

An arbitrary intersection of retraction-invariant subgroups is retraction-invariant. This follows from its being an invariance property.

An arbitrary join of retraction-invariant subgroups is retraction-invariant. This follows from its being an endo-invariance property.