Retract (group theory)

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, H is a retract of G if and only if there is an endomorphism sigma:G o G such that sigma(h) = h for all h in H and sigma(g) in H for all g in G.

The endomorphism itself is termed 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 regular factor, and in particular, every free factor, is a retract.

References


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