Retract (group theory)

Subgroup of a group in mathematics

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

The endomorphism  σ {\displaystyle \sigma } 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

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* Retraction (category theory)
* Retraction (topology)

## References

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1. ^ a b c Baer, Reinhold (1946), "Absolute retracts in group theory", Bulletin of the American Mathematical Society, 52 (6): 501–506, doi:10.1090/S0002-9904-1946-08601-2, MR 0016419.
2. ^ a b Lyndon, Roger C.; Schupp, Paul E. (2001), Combinatorial group theory, Classics in Mathematics, Berlin: Springer-Verlag, 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, vol. 2, Dordrecht: Kluwer Academic Publishers, 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, S2CID 119323021.
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, S2CID 122193204.

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