Inductive step in proof of Cauchy's theorem for finite groups

Question

The following proof was given in my set of course notes

My question is if $p$ divides $|C_{G}(g)|$ how exactly does the proof follow by induction on $|G|$?

Answer 1

Because $C_G(g) < G$, so Cauchy's theorem as an inductive hypothesis guarantees the existence of an element of order $p$ inside $C_G(g)$ and hence inside $G$ as well.