Size of Conjugacy Class is Index of Normalizer
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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: $\S 51$