Show that if \(N\trianglelefteq G\) and \(C\) is a conjugacy class in \(G\), then either \(C\subseteq N\) or \(C\cap N=\varnothing\).
Solution
Suppose \(C\cap N\ne\varnothing\), and choose \(h\in C\cap N\). Since \(C\) is the conjugacy class of \(h\), \[ C=\{ghg^{-1}:g\in G\}. \] Normality of \(N\) implies \(ghg^{-1}\in N\) for every \(g\in G\). Hence \(C\subseteq N\).
Therefore every conjugacy class is either contained in \(N\) or disjoint from \(N\).