Suppose that G is a finite abelian group Prove that G has or

Suppose that G is a finite abelian group. Prove that G has order p^n , where p  is prime, if and only if the order of every element of G  is a power of p .

Solution

If G is a power of p, then the order of any element of G is a power of p since the
order of any element divides the size of the group.
Conversely, assume all elements of G have p-power order. To show G is a power of p,
suppose it is not, so G is divisible by a prime q not equal to p. Then, by Cauchy, G has an element
of order q, and that’s a contradiction of our assumption.