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, ${\displaystyle H}$ is a retract of ${\displaystyle G}$ if and only if there is an endomorphism ${\displaystyle \sigma :G\to G}$ such that ${\displaystyle \sigma (h)=h}$ for all ${\displaystyle h\in H}$ and ${\displaystyle \sigma (g)\in H}$ for all ${\displaystyle g\in G}$.[1][2]

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

See also

• Retraction (category theory)

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 0654396.