I am reading Zee's Group theory in a nutshell for physicists and came across the following theorem (Page 96):
The all-important unitarity theorem states that finite groups have unitary representations, that is to say, $D^\dagger(g)D(g)=I$ for all $g$ and for all representations.
In practice, this theorem is a big help in finding representations of finite groups. As a start, we can eliminate some proposed representations by merely checking if the listed matrices are unitary or not.
According to me, the theorem states that all representations of a finite group are unitary. But I am having a hard time digesting this fact. The only take-away that I could draw from the proof is that given a finite group, we can always find a unitary representation for it. Can someone please explain me how the statement and the proof of the theorem are consistent with each other?
Also, a quick google search for "unitarity theorem" does not gives me anything relevant. Am I missing something? Or does this theorem have a more famous name?
I am just starting with group theory and have no background on abstract algebra so you may have to dumb it down a little for me.