character of a group is constant on each conjugacy class in the group I'm trying to solve this problem and I'm not quite sure what I need to prove here. Can you guys please help?. Here is the question.
Definition:
Character of a group $G$ is defined to be a group homomorphism  $\chi: G \rightarrow \mathbb{C^*}$. Prove that the character of a group $G$ is constant on each conjugacy class in $G$.
My answer: Let $x\in G$ be fixed. Then $\chi(gxg^{-1})=\chi(g) \chi(x) \chi(g) ^{-1}$ since $\chi$ is a homomorphism. But $\mathbb{C^*}$ is abelian so $\chi(gxg^{-1})=\chi(x)$.
Have I proved what I needed to prove? :-). I guess my question is what exactly do they mean by "constant on each conjugacy class in $G$". 
Thanks for all your help. 

You are absolutely right in your work. Conjugacy class of an element $x$ in $G$ is precisely the set $\{gxg^{-1} | g \in G\}$. Thus, you have accomplished what you wanted.
