I have this Lemma: If is a finite abelian group whose order is divisible by a prime , then contains an element of order .

I have shown that for , and order of is , then is an element in of order .

Now, suppose that has order , where . Since is abelian, is a normal subgroup of G and is an abelian group of order . It is easy to see that is divisible by (since is divisible by and ) and is less than .

Afterwards I know I need to show that there is an element in whose order is , but I'm stuck here. Please help me in completing the proof.

I have another corollary: A finite group is a -group is a power of .

I have shown .

For , I used Lagrange's Theorem and shown that every subgroup of has order of power of . How to continue the proof from here?