Suppose that and further suppose that is simple. Prove that .
Well, yes, but so is the fact that the centre is non-trivial. As the OP doesn't appear to know that the centre is non-trivial, we must assume they don't know that either! My point was, if one is going to prove one of `maximal subgroups have order p', `Z(G) is non-trivial', one would prove the latter.
Unless, of course, one doesn't know the class formula...
(I've used `one' as if I'd used `you' the last line is obviously false, because I'm pretty sure you know the class formula...so I was trying to not insult you, not trying to sound posh...)