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i have learn that intersection of pure subgroup of a group G is not necessarily pure. Can someone show me an example when such a case exists?

I'm aware that if G is torsion-free, then intersection of pure subgroup of G are necessarily pure. So the example above must involve for which the group G is not torsion-free.

Any idea? thanks

edit: The group G here we are talking about is abelian, of course.

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This is exercise 10.33(i) in Rotman's "An Introduction to Group Theory". You can take $G=\mathbb Z_2 \times \mathbb Z_8$. Take $A$ the subgroup generated by (0,1) and $B$ the subgroup generated by (1,1). Then $A$ and $B$ are pure, but their intersection will be the subgroup generated by (0,2), which is not pure in $G$.

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    @Seoral: $G^p$ is the group of $p$th powers of elements of $G$.2011-02-17