The set of all conjugation classes of group $G$ form a partition of $G$ So the property I would like to proof is the same as stated in the title:
Consider the group $G,\cdot$. The set of all conjugation classes of group $G$ form a partition of $G$
So to prove this, one needs to show that the union of conjugation classes of $G$ is equal to the group $G$. And that two different conjugation classes are disjunct. But I don't know how to proof both parts.

Hint:
You can prove this by showing that the relation $\sim$ on $G$ defined by:$$g\sim h\iff \exists x\in G\;[gx=xh]$$is an equivalence relation.
The equivalence classes (they form a partition of $G$) are exactly the conjugation classes.
