Pure subgroup

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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.


[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)
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Pure subgroup. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article