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 identity on the subgroup. In symbols, is a retract of if and only if there is an endomorphism such that for all and for all . The endomorphism itself (having this property) 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. See also Retraction (category theory) References Group theory Subgroup properties