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Retract (group theory)

Subgroup of a group in mathematics From Wikipedia, the free encyclopedia

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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 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:

  • A subgroup is a retract if and only if it has a normal complement.[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]
  • Every retract has the congruence extension property.
  • Every regular factor, and in particular, every free factor, is a retract.
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