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 the identity on the subgroup. In symbols, [math]\displaystyle{ H }[/math] is a retract of [math]\displaystyle{ G }[/math] if and only if there is an endomorphism [math]\displaystyle{ \sigma: G \to G }[/math] such that [math]\displaystyle{ \sigma(h) = h }[/math] for all [math]\displaystyle{ h \in H }[/math] and [math]\displaystyle{ \sigma(g) \in H }[/math] for all [math]\displaystyle{ g \in G }[/math].[1][2]
The endomorphism [math]\displaystyle{ \sigma }[/math] 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
References
- ↑ 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.
- ↑ 2.0 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
- ↑ 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.
- ↑ 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.
- ↑ 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.
 |  |
