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]
The endomorphism 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:
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]
The endomorphism 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: