5
$\begingroup$

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.)

$\endgroup$
2
  • 2
    $\begingroup$ Frobenius' Theorem for finite groups $\endgroup$
    – kabenyuk
    Commented May 11, 2022 at 5:48
  • $\begingroup$ Isomorphism testing of coprime extensions, where the normal Hall $N$ subgroup is Abelian. The case when $N$ is elementary Abelian is first considered; character theory is used here. Then we reduce from the case when $N$ is arbitrary Abelian to the case when $N$ is elementary Abelian. (drops.dagstuhl.de/opus/volltexte/2011/3044) $\endgroup$
    – ml0105
    Commented May 11, 2022 at 6:28

2 Answers 2

Reset to default
6
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

  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

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .