Difference between revisions of "Pure subgroup"
From Encyclopedia of Mathematics
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''serving subgroup'' | ''serving subgroup'' | ||
| − | A 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|$p$-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==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "The theory of groups" , '''1–2''' , Chelsea (1955–1956) (Translated from Russian)</TD></TR> |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)</TD></TR></table> | |
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Latest revision as of 05:49, 20 June 2023
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) |
| [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=16485
Pure subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pure_subgroup&oldid=16485
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article