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 .[1][2]
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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 normalcomplement.[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]
^For an example of a normal subgroup that is not a retract, and therefore is not a direct factor, see García, O. C.; Larrión, F. (1982), "Injectivity in varieties of groups", Algebra Universalis, 14 (3): 280–286, doi:10.1007/BF02483931, MR0654396, S2CID122193204.