Pure Subgroup - Examples

Examples

• Every direct summand of a group is a pure subgroup
• Every pure subgroup of a pure subgroup is pure.
• A divisible subgroup of an Abelian group is pure.
• If the quotient group is torsion-free, the subgroup is pure.
• The torsion subgroup of an Abelian group is pure.
• The union of pure subgroups is a pure subgroup.

Since in a finitely generated Abelian group the torsion subgroup is a direct summand, one might ask if the torsion subgroup is always a direct summand of an Abelian group. It turns out that it is not always a summand, but it is a pure subgroup. Under certain mild conditions, pure subgroups are direct summands. So, one can still recover the desired result under those conditions, as in Kulikoff's paper. Pure subgroups can be used as an intermediate property between a result on direct summands with finiteness conditions and a full result on direct summands with less restrictive finiteness conditions. Another example of this use is Prüfer's paper, where the fact that "finite torsion Abelian groups are direct sums of cyclic groups" is extended to the result that "all torsion Abelian groups of finite exponent are direct sums of cyclic groups" via an intermediate consideration of pure subgroups.