Definition of Conjugacy Class
Solution 1:
By definition, $C_a = \{gag^{-1}:g \in G\}$, so given any $h \in G$, $$h \Omega_a h^{-1} = \sum_{g \in C_a} hgh^{-1}$$ But $g$ is obtained by conjugating $a$, so $hgh^{-1}$ is also some conjugation of $a$. It is then clear that we are simply permuting the sum, so $$h \Omega_a h^{-1} = \Omega_a$$