# Сondition for the existence of the subgroup

I have an abelian group $G$ of order $m$. And I want to know if there is any subgroup $H$ with order $n$. The condition of the Lagrange's theorem ($m = 0\ (mod\ n)$) seems to be necessary but insufficient.

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Do you know that finite abelian groups can be written as a direct product? –  mixedmath Sep 12 '11 at 13:24
What examples have you seen of insufficiency? –  lhf Sep 12 '11 at 13:36
@lhf, I didn't find any. So as a proof of insufficient. –  Ximik Sep 12 '11 at 13:38
• Every finite abelian $p$-group is a product of cyclic subgroups and so the converse of Lagrange's theorem holds for abelian $p$-groups.