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, H is a retract of G if and only if there is an endomorphism 
such that σ(h) = h for all 
and 
for all 
.
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.
