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A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263701.png" /> of a (partially) [[Ordered group|ordered group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263702.png" /> which is a [[Convex subset|convex subset]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263703.png" /> with respect to the given order relation. Normal convex subgroups are exactly the kernels of homomorphisms of the partially ordered group which preserve the order. A subgroup of an [[Orderable group|orderable group]] which is convex for any total order is called an absolutely convex subgroup; if it is convex only for a certain total order, it is called a relatively convex subgroup. The intersection of all non-trivial relatively convex subgroups of an orderable group is an absolutely convex subgroup; the union of all proper relatively convex subgroups is also an absolutely convex subgroup. Torsion-free Abelian groups have no non-trivial absolutely convex subgroups. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263704.png" /> of a completely ordered group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263705.png" /> is absolutely convex if and only if for any elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263707.png" /> the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263708.png" /> is non-empty, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c0263709.png" /> is the minimal invariant sub-semi-group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c02637010.png" /> containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c02637011.png" />. A convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c02637012.png" />-subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c02637013.png" /> of a [[Lattice-ordered group|lattice-ordered group]] is isolated, i.e. for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c02637014.png" />, it follows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c02637015.png" /> that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026370/c02637016.png" />.
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A subgroup $H$ of a (partially) [[Ordered group|ordered group]] $G$ which is a [[Convex subset|convex subset]] of $G$ with respect to the given order relation. Normal convex subgroups are exactly the kernels of homomorphisms of the partially ordered group which preserve the order. A subgroup of an [[Orderable group|orderable group]] which is convex for any total order is called an absolutely convex subgroup; if it is convex only for a certain total order, it is called a relatively convex subgroup. The intersection of all non-trivial relatively convex subgroups of an orderable group is an absolutely convex subgroup; the union of all proper relatively convex subgroups is also an absolutely convex subgroup. Torsion-free Abelian groups have no non-trivial absolutely convex subgroups. A subgroup $H$ of a completely ordered group $G$ is absolutely convex if and only if for any elements $g\not\in H$, $a\in H$ the intersection $S(g)\cap S(ga)$ is non-empty, where $S(x)$ is the minimal invariant sub-semi-group of $G$ containing $x$. A convex $l$-subgroup $H$ of a [[Lattice-ordered group|lattice-ordered group]] is isolated, i.e. for any natural number $n$, it follows from $x^n\in H$ that $x\in H$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Kokorin,  V.M. Kopytov,  "Fully ordered groups" , Israel Program Sci. Transl.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Kokorin,  V.M. Kopytov,  "Fully ordered groups" , Israel Program Sci. Transl.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L. Fuchs,  "Partially ordered algebraic systems" , Pergamon  (1963)</TD></TR></table>

Latest revision as of 16:16, 12 April 2014

A subgroup $H$ of a (partially) ordered group $G$ which is a convex subset of $G$ with respect to the given order relation. Normal convex subgroups are exactly the kernels of homomorphisms of the partially ordered group which preserve the order. A subgroup of an orderable group which is convex for any total order is called an absolutely convex subgroup; if it is convex only for a certain total order, it is called a relatively convex subgroup. The intersection of all non-trivial relatively convex subgroups of an orderable group is an absolutely convex subgroup; the union of all proper relatively convex subgroups is also an absolutely convex subgroup. Torsion-free Abelian groups have no non-trivial absolutely convex subgroups. A subgroup $H$ of a completely ordered group $G$ is absolutely convex if and only if for any elements $g\not\in H$, $a\in H$ the intersection $S(g)\cap S(ga)$ is non-empty, where $S(x)$ is the minimal invariant sub-semi-group of $G$ containing $x$. A convex $l$-subgroup $H$ of a lattice-ordered group is isolated, i.e. for any natural number $n$, it follows from $x^n\in H$ that $x\in H$.

References

[1] A.I. Kokorin, V.M. Kopytov, "Fully ordered groups" , Israel Program Sci. Transl. (1974) (Translated from Russian)
[2] L. Fuchs, "Partially ordered algebraic systems" , Pergamon (1963)
How to Cite This Entry:
Convex subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_subgroup&oldid=31640
This article was adapted from an original article by A.I. Kokorin, V.M. Kopytov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article