About: Retract (group theory)

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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:

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  • 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 , and in particular, every , is a retract. (en)
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  • 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: (en)
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  • Retract (group theory) (en)
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