# Example of Intersection of Pure Subgroup which is not Pure

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 $G$.
Does the subgroup $A$ looks like $<(1,0)>=\{(0,0),(1,0)\}$ and $B$ looks like $<(1,1)>=\{(1,1),(0,2),(1,3),(0,4),(1,5),(0,6),(1,7),(0,0)\}.$ Then the intersection of their subgroup will not be generated by (0,2) or do you mean that $A$ is generated by $(0,1)$? – Seoral Jan 25 '11 at 5:31
Jack: You mention that their intersection is contained in $G^p$... what is $G^p$? Sorry for the late revert. I just come back from chinese new year. Cheers. – Seoral Feb 9 '11 at 7:04