Retract (group theory)

In mathematics, in the field of group theory, a subgroup of a group is termed a retract if there is an endomorphism of the group that maps surjectively to the subgroup and is the identity on the subgroup. In symbols, $$H$$ is a retract of $$G$$ if and only if there is an endomorphism $$\sigma: G \to G$$ such that $$\sigma(h) = h$$ for all $$h \in H$$ and $$\sigma(g) \in H$$ for all $$g \in G$$.

The endomorphism $$\sigma$$ is an idempotent element in the transformation monoid of endomorphisms, so it is called an idempotent endomorphism or a retraction.

The following is known about retracts:


 * A subgroup is a retract if and only if it has a normal complement. The normal complement, specifically, is the kernel of the retraction.
 * Every direct factor is a retract. Conversely, any retract which is a normal subgroup is a direct factor.
 * Every retract has the congruence extension property.
 * Every regular factor, and in particular, every free factor, is a retract.