Let

be a group of order

, with

and

(not necessarily distinct) primes. Prove that the center

or

.

I started off by doing a proof by contradiction.

Suppose that

,

Then

divides either

or

, which implies that

or

We know that

is a normal subgroup.

Consider the quotient group

.

Using Larange's Theorem, we know that

(*).

I broke it into two cases: 1:

, 2:

.

If under case 1, (*) evaluates to

. Otherwise, in case 2, (*) evaluates to

.

This is as far as I've gotten and I'm pretty stumped as how else to follow up after.