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 identity on the subgroup. In symbols, formula_1 is a retract of formula_2 if and only if there is an endomorphism formula_3 such that formula_4 for all formula_5 and formula_6 for all formula_7.
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: