Lemma. Suppose $G$ is a group and $\emptyset \neq S \subset G$. Set $N=N_G(S)$. Let $\{g_i\mid i \in I\}$ be a complete set of right cosets representations for $N$ in $G$. Then the conjugates of $S$ in $G$ are $\{S^{g_i}\mid i \in I\}$ and $S^{g_i}=S^{g_j}$ iff $g_i=g_j$.
What is a complete set of right cosets representations? I just don't get what they actually are. I know you can get an equivalence class out of the conjugate classes.
Also, can someone explain what the class equation of finite group is?
As by the definition he has. Let $G$ be a finite group. Let $x_1,x_2,...,x_k$ be elements of $G$ one from each of the $R$ conjugacy classes.
Set $n_i=|x_i^G|$. Assume notation chosen so as $n_1=...=n_l$ and $n_i>1$ for $i>l$
i) $|G|=\sum_{i=1}^k n_i= \sum_{i=1}^k[G:C_{G}(x_i)]$