Application of character theory to structure of groups

Question

One application of character theory in the investigation of structure of finite groups is for Burnside's theorem.

Can one mention some other results in Group theory, whose proofs are elementary from character theory, but difficult without character theory?

(There was a similar question on mathstack, but I did not find many answers in the direction above.)

Answer 1

If you do not wish to use character theory or calculations in a group ring for that matter, try this one: for a finite group $G$, let $g \in G$ be a commutator (that is, $g=[x,y]=x^{-1}y^{-1}xy$ for certain $x,y \in G$) and $n \in \mathbb{Z}$, such that gcd$(n, o(g))=1$ (here $o(g)$ denotes the order of $g$). Then $g^n$ is again a commutator.

Answer 2

1. Let $G$ be a finite group with an abelian Sylow $p$-subgroup. Then $|G'\cap Z(G)|$ is not divisible by $p$.

2. Let $g\in G$ and $\overline{G}=G/N$. Then $|C_{\overline{G}}(\overline{g})|\le |C_G(g)|$.

As I know, it is "easier" to proof the first result using a character theory (the key concept is a frobenius reciprocity), while a purely group-theoretic approach uses a transfer map (that is, of course, related to induced characters, but still).

The second orthogonality relation easily deals with the second result, while calculations using group methods are tedious