Why, for all $i$, $Z(G)\neq \bar x_i$

Solution 1:

If $x$ is any element of $G$, then the conjugacy class $\overline{x} = \{ gxg^{-1} : g\in G\}$ is the singleton set $\{x\}$ if and only if $x$ lies in the center of $G$. By definition, the elements $x_1, ... , x_k$ in your post should be chosen to not lie in the center of $G$. They are representatives of the conjugacy classes which are not singleton sets.

Thus $Z(G)$ is the disjoint union of the conjugacy classes which each consist of only one element, and $\overline{x_1}, ... , \overline{x_k}$ are the remaining conjugacy classes, each of which contains more than one element.