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, is a retract of if and only if there is an endomorphism such that for all and for all .
The endomorphism 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.For an example of a normal subgroup that is not a retract, and therefore is not a direct factor, see . Every retract has the congruence extension property. Every regular factor, and in particular, every free factor, is a retract.
See also Retraction (category theory) Retraction (topology)
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