let G be a nonabelian group of odd order. I am searching a proof of the fact that then every non-central conjugacy class size occurs at least twice.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
1
Hint: show that in a group of odd order no element can be conjugate to its inverse. Indeed, if $x^{-1} = gxg^{-1}$, then what is $g^nxg^{-n}$?
See if you can deduce the claim from this observation.
-
1$\begingroup$ Hi, thank you very much for your hint. I think, I've got it now: We assume that $x^{-1}=gxg^{-1}$. It follows $x=gx^{-1}g^{-1}$ (because $xx^{-1}=e=x^{-1}x$). Therefore, $x^{-1}=gxg^{-1}=g^2x^{-1}g^{-2}$ and iterating this yields $x^{-1}=g^{o(g)}xg^{-o(g)}=x$. But $x^2=e$ is impossible, because $G$ has odd order. Therefore, class$_G(x)\neq$ class$_G(x^{-1})$ and class$_G(x)\cap$ class$_G(x^{-1})=\emptyset$. We now have $|$class$_G(x)|=\frac{|G|}{|C_G(x)|}=\frac{|G|}{|C_G(x^{-1})|}=|$class$_G(x^{-1})|$, and the statement is proven. $\endgroup$ Oct 30, 2011 at 19:07