Retract (group theory)

Short description: 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 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.[1][2]

The endomorphism σ 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)
* Retraction (topology)

## References

1. ↑ 1.0 1.1 1.2 "Absolute retracts in group theory", Bulletin of the American Mathematical Society 52 (6): 501–506, 1946, doi:10.1090/S0002-9904-1946-08601-2 .
2. ↑ 2.0 2.1 Lyndon, Roger C.; Schupp, Paul E. (2001), Combinatorial group theory, Classics in Mathematics, Berlin: Springer-Verlag, p. 2, ISBN 3-540-41158-5, https://books.google.com/books?id=aiPVBygHi_oC&pg=PA2
3. ↑ Krylov, Piotr A.; Mikhalev, Alexander V.; Tuganbaev, Askar A. (2003), Endomorphism rings of abelian groups, Algebras and Applications, 2, Dordrecht: Kluwer Academic Publishers, p. 24, doi:10.1007/978-94-017-0345-1, ISBN 1-4020-1438-4, https://books.google.com/books?id=iy4sVSgzrvYC&pg=PA24 .
4. ↑ Myasnikov, Alexei G.; Roman'kov, Vitaly (2014), "Verbally closed subgroups of free groups", Journal of Group Theory 17 (1): 29–40, doi:10.1515/jgt-2013-0034 .
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 .

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Retract (group theory). Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Retract_(group_theory)&oldid=193065"