# Pure subgroup

*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