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 $\sigma:G \to G$ such that $\sigma(h) = h$ for all $h \in H$ and $\sigma(g) \in H$ for all $g \in G$.^[1][2]

The endomorphism itself (having this property) is an idempotent element in the transformation monoid of endomorphisms, so it 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.

References

1. Baer, Reinhold (1946), "Absolute retracts in group theory", Bulletin of the American Mathematical Society 52: 501–506, doi:10.1090/S0002-9904-1946-08601-2, MR 0016419.
2. Lyndon, Roger C.; Schupp, Paul E. (2001), Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, p. 2, ISBN 3-540-41158-5, MR 1812024.
3. Krylov, Piotr A.; Mikhalev, Alexander V.; Tuganbaev, Askar A. (2003), Endomorphism rings of abelian groups, Algebras and Applications 2, Kluwer Academic Publishers, Dordrecht, p. 24, doi:10.1007/978-94-017-0345-1, ISBN 1-4020-1438-4, MR 2013936.
4. Myasnikov, Alexei G.; Roman'kov, Vitaly (2014), "Verbally closed subgroups of free groups", Journal of Group Theory 17 (1): 29–40, arXiv:1201.0497, doi:10.1515/jgt-2013-0034, MR 3176650.
5. For an example of a normal subgroup that is not a retract, and therefore is not a direct factor, see García, O. C.; Larrión, F. (1982), "Injectivity in varieties of groups", Algebra Universalis 14 (3): 280–286, doi:10.1007/BF02483931, MR 654396.