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Difference between revisions of "Pure subgroup"

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''serving subgroup''
 
''serving subgroup''
  
A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759001.png" /> of an [[Abelian group|Abelian group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759002.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759003.png" /> the solvability of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759004.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759005.png" /> implies its solvability in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759006.png" />. Examples of pure subgroups are the zero subgroup, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759007.png" /> itself, the torsion part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759008.png" />, and direct summands. Not every pure subgroup need be a direct summand, even for a [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p0759009.png" />-group]]. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p07590010.png" /> is a torsion pure subgroup of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p07590011.png" /> and if the orders of its elements are uniformly bounded, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p07590012.png" /> is a direct summand in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075900/p07590013.png" />. There is a complete description of the Abelian groups in which every pure subgroup is a direct summand (see [[#References|[1]]]). The question of the cardinality of the set of pure subgroups of an Abelian group has been thoroughly investigated.
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A subgroup $C$ of an [[Abelian group|Abelian group]] $G$ such that for any $c\in G$ the solvability of the equation $nx=c$ in $G$ implies its solvability in $C$. Examples of pure subgroups are the zero subgroup, $G$ itself, the torsion part of $G$, and direct summands. Not every pure subgroup need be a direct summand, even for a [[P-group|$p$-group]]. However, if $C$ is a torsion pure subgroup of an Abelian group $G$ and if the orders of its elements are uniformly bounded, then $C$ is a direct summand in $G$. There is a complete description of the Abelian groups in which every pure subgroup is a direct summand (see [[#References|[1]]]). The question of the cardinality of the set of pure subgroups of an Abelian group has been thoroughly investigated.
  
 
====References====
 
====References====

Revision as of 08:30, 1 August 2014

serving subgroup

A subgroup $C$ of an Abelian group $G$ such that for any $c\in G$ the solvability of the equation $nx=c$ in $G$ implies its solvability in $C$. Examples of pure subgroups are the zero subgroup, $G$ itself, the torsion part of $G$, and direct summands. Not every pure subgroup need be a direct summand, even for a $p$-group. However, if $C$ is a torsion pure subgroup of an Abelian group $G$ and if the orders of its elements are uniformly bounded, then $C$ is a direct summand in $G$. There is a complete description of the Abelian groups in which every pure subgroup is a direct summand (see [1]). The question of the cardinality of the set of pure subgroups of an Abelian group has been thoroughly investigated.

References

[1] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)


Comments

References

[a1] D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
How to Cite This Entry:
Pure subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pure_subgroup&oldid=32632
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article