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, ${\displaystyle H}$is a retract of ${\displaystyle G}$if and only if there is an endomorphism ${\displaystyle \sigma :G\to G}$such that ${\displaystyle \sigma (h)=h}$for all ${\displaystyle h\in H}$and ${\displaystyle \sigma (g)\in H}$for all ${\displaystyle g\in G}$.

The endomorphism ${\displaystyle \sigma }$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.

• Retraction (category theory)
• Retraction (topology)