# Math Help - finite abelian groups

1. ## finite abelian groups

G finite abelian group
1 subgroup of order d for each divisor d of |G|.
Prove G cyclic.

2. Originally Posted by stumped765
G finite abelian group
1 subgroup of order d for each divisor d of |G|.
Prove G cyclic.
What if we proved the contrapositive. Suppose that $G$ is not cyclic. Try constructing subgroups of $G$ which are distinct but have the same order (hint: you're going to want to construct these subgroups to be cyclic groups [math}C_n,C_m[/tex] say with some prime $p\mid m,n$)