Retract (group theory) explained

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,

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

The endomorphism

\sigma is an idempotent element in the transformation monoid of endomorphisms, so it is called an idempotent endomorphism^{[1] [2]} or a retraction.

The following is known about retracts:

• A subgroup is a retract if and only if it has a normal complement.^[3] 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.^[4]
• Every retract has the congruence extension property.
• Every regular factor, and in particular, every free factor, is a retract.

See also

• Retraction (category theory)
• Retraction (topology)

Notes and References

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3. .
4. For an example of a normal subgroup that is not a retract, and therefore is not a direct factor, see .