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.