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I want to find the sufficient and necessary condition of intersection of maximal subgroup of abelian group is not trivial.

I showed intersection of maximal subgroup of a group $G$ is normal subgroup of $G$. I know fundamental theorem of abelian group.

Thank you for your help.

Shaun
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Poitou-Tate
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    The fundamental theorem only applies to finitely generated abelian groups. In that case a necessary and sufficient condition is the existence of an element of order $p^2$ for some prime $p$. – Derek Holt Nov 08 '20 at 12:41
  • Thank you very much. How can I prove that? – Poitou-Tate Nov 08 '20 at 13:03
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    @bellow: Use the fundamental theorem; an element of order $p^2$ means a factor of the form $C_{p^n}$ with $n\gt 2$. Show every maximal subgroup includes at least the maximum subgroup of that factor. – Arturo Magidin Nov 08 '20 at 23:24

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