The Order of a Conjugacy Class Divides the Order of the Group

Group Theory Problems and Solutions in Mathematics

Problem 455

Let $G$ be a finite group.
The centralizer of an element $a$ of $G$ is defined to be
\[C_G(a)=\{g\in G \mid ga=ag\}.\]

A conjugacy class is a set of the form
\[\Cl(a)=\{bab^{-1} \mid b\in G\}\] for some $a\in G$.


(a) Prove that the centralizer of an element of $a$ in $G$ is a subgroup of the group $G$.

(b) Prove that the order (the number of elements) of every conjugacy class in $G$ divides the order of the group $G$.

 
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Proof.

(a) Prove that the centralizer of $a$ in $G$ is a subgroup of $G$.

Since the identity element $e$ of $G$ satisfies $ea=a=ae$, it is in the centralizer $C_G(a)$.
Hence $C_G(a)$ is not an empty set. We show that $C_G(a)$ is closed under multiplications and inverses.

Let $g, h \in C_G(a)$. Then we have
\begin{align*}
(gh)a&=g(ha)\\
&=g(ah) && \text{since $h\in C_G(a)$}\\
&=(ga)h\\
&=(ag)h&& \text{since $g\in C_G(a)$}\\
&=a(gh).
\end{align*}
So $gh$ commutes with $a$ and thus $gh \in C_G(a)$.
Thus $C_G(a)$ is closed under multiplications.

Let $g\in C_G(a)$. This means that we have $ga=ag$.
Multiplying by $g^{-1}$ on the left and on the right, we obtain
\begin{align*}
g^{-1}(ga)g^{-1}=g^{-1}(ag)g^{-1},
\end{align*}
and thus we have
\[ag^{-1}=g^{-1}a.\] This implies that $g^{-1}\in C_G(a)$, hence $C_G(a)$ is closed under inverses.

Therefore, $C_G(a)$ is a subgroup of $G$.

(b) Prove that the order of every conjugacy class in $G$ divides the order of $G$.

We give two proofs for part (b).The first one is a more direct proof and the second one uses the orbit-stabilizer theorem.

The First Proof of (b).

By part (a), the centralizer $C_G(a)$ is a subgroup of the finite group $G$.
Hence the set of left cosets $G/C_G(a)$ is a finite set, and its order divides the order of $G$ by Lagrange’s theorem.

We prove that there is a bijective map from $G/C_G(a)$ to $\Cl(a)$.
Define the map $\phi:G/C_G(a) \to \Cl(a)$ by
\[\phi\left(\, gC_G(a) \,\right)=gag^{-1}.\]

We must show that it is well-defined.
For this, note that we have
\begin{align*}
gC_G(a)=hC_G(a) &\Leftrightarrow h^{-1}g\in C_G(a)\\
& \Leftrightarrow (h^{-1}g)a(h^{-1}g)^{-1}=a\\
& \Leftrightarrow gag^{-1}=hag^{-1}.
\end{align*}
This computation shows that the map $\phi$ is well-defined as well as $\phi$ is injective.
Since the both sets are finite sets, this implies that $\phi$ is bijective.
Thus, the order of the two sets is equal.

It yields that the order of $C_G(a)$ divides the order of the finite group $G$.

The Second Proof of (b). Use the Orbit-Stabilizer Theorem

We now move on to the alternative proof.
Consider the action of the group $G$ on itself by conjugation:
\[\psi:G\times G \to G, \quad (g,h)\mapsto g\cdot h=ghg^{-1}.\]

Then the orbit $\calO(a)$ of an element $a\in G$ under this action is
\[\calO(a)=\{ g\cdot a \mid g\in G\}=\{gag^{-1} \mid g\in G\}=\Cl(a).\]

Let $G_a$ be the stabilizer of $a$.
Then the orbit-stabilizer theorem for finite groups say that we have
\begin{align*}
|\Cl(a)|=|\calO(a)|=[G:G_a]=\frac{|G|}{|G_a|}
\end{align*}
and hence the order of $\Cl(a)$ divides the order of $G$.

Note that the stabilizer $G_a$ of $a$ is the centralizer $C_G(a)$ of $a$ since
\[G_a=\{g \in G \mid g\cdot a =a\}=\{g\in G \mid ga=ag\}=C_G(a).\]


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