Size of Conjugacy Class is Index of Normalizer

Theorem

Let $G$ be a group.

Let $x \in G$.

Let $\conjclass x$ be the conjugacy class of $x$ in $G$.

Let $\map {N_G} x$ be the normalizer of $x$ in $G$.

Let $\index G {\map {N_G} x}$ be the index of $\map {N_G} x$ in $G$.

The number of elements in $\conjclass x$ is $\index G {\map {N_G} x}$.

Proof

The number of elements in $\conjclass x$ is the number of conjugates of the set $\set x$.

From Number of Distinct Conjugate Subsets is Index of Normalizer, the number of distinct subsets of a $G$ which are conjugates of $S \subseteq G$ is $\index G {\map {N_G} S}$.

The result follows.

$\blacksquare$

Sources

• 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): $1$: Subgroups: $1. \text{T}.3$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 51$