Use Cauchy’s Theorem and induction to prove that a finite abelian group $G$ has a subgroup of order $d$ for all $d$ dividing the order of $G$.
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1) From Cauchy's Theorem for any prime dividing the group's order there's an element $\,x_p\,$ of order $\,p\,$
2) The claim follows at once for $\,|G|=1,2,3,4\,$ and, in fact, for every prime.
3) Use (1) above and the fact that every subgroup of an abelian group is normal and look at the group $\,G/\langle x_p\rangle\,$
4) Apply the inductive hypothesis to the group in (3) and end the argument.