If a finite abelian group has order a power of a prime p,
then the order of every element in the group is a power of p.
I used Lagrange's theorem that order of element in Group (order of subgroup generated by element) must divide order of the group.
If G is order of power of prime, then order of element is power of prime.
But I can't understand whether this group is cyclic or not.
Also, Using Lagrange's theorem does not tell me whether every element has power of prime.
I think order of element does not necessarily have to be power of prime, as long as order of element divides power of prime.