Suppose G is a finite abelian p-group. Prove that if G is cyclic iff G has exactly p-1 elements of order p.

$\displaystyle \forall g\in G, \ \ g^{p^{\alpha}}=1, \ \ |G|=p^{\alpha}$

$\displaystyle (\Rightarrow)$

Suppose G is cyclic. Then for a $\displaystyle g\in G$, $\displaystyle G=<g>$.

What next? I am at a loss.