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 $\backslash sigma:\; G\; \backslash to\; G$ such that $\backslash sigma(h)\; =\; h$ for all $h\; \backslash in\; H$ and $\backslash sigma(g)\; \backslash in\; H$ for all $g\; \backslash in\; G$.{{citation

| last1 = Lyndon | first1 = Roger C.

| author-link1= Roger Lyndon

| last2 = Schupp | first2 = Paul E.

| authorlink2= Paul Schupp

| isbn = 3-540-41158-5

| mr = 1812024

| page = 2

| publisher = Springer-Verlag | location = Berlin

| series = Classics in Mathematics

| title = Combinatorial group theory

| url = https://books.google.com/books?id=aiPVBygHi_oC&pg=PA2

| year = 2001}}

The endomorphism $\backslash sigma$ is an idempotent element in the transformation monoid of endomorphisms, so it is called an idempotent endomorphism{{citation

| last = Baer | first = Reinhold | authorlink = Reinhold Baer

| doi = 10.1090/S0002-9904-1946-08601-2

| journal = Bulletin of the American Mathematical Society

| mr = 0016419

| pages = 501–506

| title = Absolute retracts in group theory

| volume = 52

| year = 1946| issue = 6 | doi-access = free

}}.{{citation

| last1 = Krylov | first1 = Piotr A.

| last2 = Mikhalev | first2 = Alexander V.

| last3 = Tuganbaev | first3 = Askar A.

| doi = 10.1007/978-94-017-0345-1

| isbn = 1-4020-1438-4

| mr = 2013936

| page = 24

| publisher = Kluwer Academic Publishers

| location = Dordrecht

| series = Algebras and Applications

| title = Endomorphism rings of abelian groups

| url = https://books.google.com/books?id=iy4sVSgzrvYC&pg=PA24

| volume = 2

| year = 2003}}. or a retraction.

The following is known about retracts:

• A subgroup is a retract if and only if it has a normal complement.{{citation

| last1 = Myasnikov | first1 = Alexei G.

| last2 = Roman'kov | first2 = Vitaly

| arxiv = 1201.0497

| doi = 10.1515/jgt-2013-0034

| issue = 1

| journal = Journal of Group Theory

| mr = 3176650

| pages = 29–40

| title = Verbally closed subgroups of free groups

| volume = 17

| year = 2014| s2cid = 119323021

}}. 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.For an example of a normal subgroup that is not a retract, and therefore is not a direct factor, see {{citation

| last1 = García | first1 = O. C.

| last2 = Larrión | first2 = F.

| doi = 10.1007/BF02483931

| issue = 3

| journal = Algebra Universalis

| mr = 654396

| pages = 280–286

| title = Injectivity in varieties of groups

| volume = 14

| year = 1982| s2cid = 122193204

}}.

• Every retract has the congruence extension property.
• Every regular factor, and in particular, every free factor, is a retract.