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, is a retract of if and only if there is an endomorphism such that for all and for all .
The following is known about retracts:
- A subgroup is a retract if and only if it has a normal complement. The normal complement, specifically, is the kernel of the retraction.
- Every direct factor is a retract. Conversely, any retract which is a normal subgroup is a direct factor.
- Every retract has the congruence extension property.
- Every regular factor, and in particular, every free factor, is a retract.
- 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.
- 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.
- 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.
- Myasnikov, Alexei G.; Roman'kov, Vitaly (2014), "Verbally closed subgroups of free groups", Journal of Group Theory, 17 (1): 29–40, arXiv: , doi:10.1515/jgt-2013-0034, MR 3176650.
- 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.
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