Conjugacy class splitting in a prime-index normal subgroup
= Conjugacy class splitting in a prime-index normal subgroup
If $N\trianglelefteq G$ has prime index $p$, a <conjugacy class> of $G$ lying in $N$ splits into one or $p$ <conjugacy classes> of $N$. <Conjugation> by $G$ acts transitively on these smaller classes, while $N$ fixes every one. This factors through $G/N$, and the <orbit-stabilizer theorem> makes the number of classes a divisor of $p$.