I would like to proof the following in the case of groups of finite order:
the number of elements of an arbitrary class of conjugated elements is a divider of the index (G:Z) of the centre of G.
Does anybody know how to proof this?
Peter Mulder
I would like to proof the following in the case of groups of finite order:
the number of elements of an arbitrary class of conjugated elements is a divider of the index (G:Z) of the centre of G.
Does anybody know how to proof this?
Peter Mulder
Let a group $\displaystyle G$ act on itself by conjugation: $\displaystyle G\times G\rightarrow G\,,\,g\cdot x\to gxg^{-1}$ , then $\displaystyle |Orb(x)|=[G:Stab(x)]$ ,
with $\displaystyle Stab(x):=\{g\in G/\,gxg^{-1}=x\}=C_G(x)=$ the centralizer of $\displaystyle x$ in $\displaystyle G$
Since $\displaystyle Orb(x)$ is the equivalence class of $\displaystyle x$ and since $\displaystyle Z(G)\leq C_G(x)$ , we're done by Lagrange's theorem
Tonio