# Let $G$ be a finite group and $H$ a normal subgroup of $G$, $[G:H] = p$ is prime and $a \in H$.

In my group theory course, I came across the following lemma which confuses me:

Let $G$ be a finite group and $H$ a normal subgroup of $G$, $[G:H] =p$ is prime and $a \in H$. Then one of two cases applies:

1) $C_H(a) \neq C_G(a)$. This means that the conjugacy class of $a$ in $H$ is the same as the conjugacy class of $a$ in $G$.

2) $C_H(a) = C_G(a)$. This means that the conjugacy class of $a$ in $G$ is a disjoint union of $p$ conjugacy classes of $a$ in $H$.

What does it mean in case 2, when it says that "This means that the conjugacy class of $a \in H$ is a disjoint union of $p$ conjugacy classes of $a$ in $H$"?

I was under the impression that the there is only one conjugacy class for each element. So why does it say the union of $p$ conjugacy classes of $a$?

• are you sure it doesn't just say it is the disjoint union of $p$ conjugacy classes? – Jorge Fernández Hidalgo Dec 14 '16 at 6:19
• @JorgeFernándezHidalgo Unless I made a mistake in writing down my notes ... hmm – MathMajor Dec 14 '16 at 6:23
• because that would be true – Jorge Fernández Hidalgo Dec 14 '16 at 6:25