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
σ
:
G
→
G
such that
σ
(
h
)
=
h
for all
h
∈
H
and
σ
(
g
)
∈
H
for all
g
∈
G
.
The endomorphism itself (having this property) is an idempotent element in the transformation monoid of endomorphisms, so it 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 regular factor, and in particular, every free factor, is a retract.
