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 ${\displaystyle g\in G}$.^[1][2]

The endomorphism ${\displaystyle \sigma }$ is an idempotent element in the transformation monoid of endomorphisms, so it is called an idempotent endomorphism^[1][3] or a retraction.^[2]

The following is known about retracts:

• A subgroup is a retract if and only if it has a normal complement.^[4] The normal complement, specifically, is the kernel of the retraction.
• Every direct factor is a retract.^[1] Conversely, any retract which is a normal subgroup is a direct factor.^[5]
• Every retract has the congruence extension property.
• Every regular factor, and in particular, every free factor, is a retract.