# Does element of the same conjugacy class commute?

Let $G$ a group and $C_1,...,C_n$ it's conjugacy class. I have to compute $[G,G]=\left<ghg^{-1}h^{-1}\mid g,h\in G\right>$. I would like to know if elements of $C_i$ commute. I know that element of $C_i$ will be equal in $G/[G,G]$, so, may be, If I'm lucky, if $A,B\in C_i$, then $[A,B]=1$. I tryed to make a proof, but not conclusive. Any idea ?

## 2 Answers

They do not. Remember all transpositions in $S_n$ are conjugate, notice $(1,2)(2,3) \neq (2,3)(1,2)$

• It's true, but $[(12)(23),(23)(12)]=1$... may be my question is not very clear. But do we always have $[A,B]=1$ if $A$ and $B$ are in the same conjugacy class ?
– MSE
Commented Jun 14, 2016 at 14:01

This is not true in general. For example, in the dihedral group $D_{2n}=\langle a,b\vert a^n=b^2=1,b^{-1}ab=a^{-1}\rangle$ for odd $n$, $b$ is conjugate with $ab,\ldots,a^{n-1}b$. Suppose that $[b,a^ib]=1$ for $1\le i \le n-1$. Then $ba^ibb^{-1}b^{-1}a^{-i}=1$. So, $ba^ib^{-1}a^{-i}=1$ and we have $b^{-1}a^ib=a^i$, that is a contradiction with the second relation of the group, $b^{-1}ab=a^{-1}$.