📂Abstract Algebra

Conjugacy Class

Definition¹

For elements $a$ and $b$ of a group $G$, we say they are conjugate if there exists some $x \in G$ such that $xax^{-1} = b$ holds. Alternatively, $b$ is called a conjugate of $a$.

The set of all conjugates of $a$ is called the conjugacy class and is denoted by $\operatorname{cl}(a)$. $$ \operatorname{cl}(a) := \left\{ xax^{-1} : x \in G \right\} $$

Remarks

For the identity element $e \in G$, $e a e^{-1} = a$ always holds, so a conjugacy class always contains the element itself. In other words, every conjugacy class is not the empty set.

$$ a \in \operatorname{cl}(a) $$

Theorem

(a) Conjugacy is an equivalence relation on $G$.

(b) For a finite group $G$ and $a \in G$, the following holds. $$ |\operatorname{cl}(a)| = |G : C(a)| $$ Here $C(a)$ is the centralizer, and $|G : C(a)|$ is the index.

(b’)

$|\operatorname{cl}(a)|$ is a divisor of $|G|$.

$$ |\operatorname{cl}(a)| = |G| / |C(a)| $$

Proof

(a)

Let $a$ and $b$ satisfying $x a x^{-1} = b$ be represented by the ordered pair $(a,b)$, and denote it by $R = \left\{ (a,b) : xax^{-1} = b \right\}$.

An equivalence relation $R$ on the group $G$ is a relation satisfying the following.

1. $(a, a) \in R$ $\forall a \in G$
2. $(a, b) \in R \implies (b, a) \in R$
3. $(a, b) \in R$, $(b, c) \in R \implies (a, c) \in R$

1. Since $eae^{-1} = a$ holds for the identity $e$ and every $a \in G$, $(a, a) \in R$ holds.

2. If $(a, b) \in R$ then $xax^{-1} = b$ holds. If we set $y = x^{-1}$, then the following holds. $$ yby^{-1} = y(xax^{-1})y^{-1} = x^{-1}xaxx^{-1} = a \implies (b,a) \in R $$

3. If $(a, b) \in R$ and $(b, c) \in R$ then $xax^{-1} = b$ and $yby^{-1} = c$ hold. If we set $z = yx$, then the following holds. $$ c = yby^{-1} = y(xax^{-1})y^{-1} = (yx)a(x^{-1}y^{-1}) = (yx) a (yx)^{-1} = zaz^{-1} $$ $$ \implies (a,c) \in R $$

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(b)

We prove this by showing that the following function is well-defined and bijective. For a fixed $a \in G$,

$$ \begin{align*} \phi : G/C(a) &\to \operatorname{cl}(a) \\ xC(a) &\mapsto xax^{-1} \end{align*} $$

Here $G/C(a)$ is the quotient group.

$\phi$ is well-defined

For every $x \in G$, the function value $\phi(xC(a)) = xax^{-1}$ exists. Also, by the properties of cosets, the following holds.

$$ \begin{align*} x_{1}C(a) = x_{2}C(a) &\iff x_{1}x_{2}^{-1} \in C(a) \\ &\iff x_{1}^{-1}x_{2}a = ax_{1}^{-1}x_{2} \\ &\iff x_{1}^{-1}x_{2}ax_{2}^{-1} = ax_{1}^{-1} \\ &\iff x_{2}ax_{2}^{-1} = x_{1}ax_{1}^{-1} \\ \end{align*} $$

Therefore $\phi$ is well-defined.

$\phi$ is injective

That $\phi$ is an injective function was shown in the equations above.

$\phi$ is surjective

For every $xax^{-1} \in \operatorname{cl}(a)$, there exists $xC(a) \in G/C(a)$. Thus $\phi$ is a surjection.

Conclusion

Since the function $\phi : G/C(a) \to \operatorname{cl}(a)$ is bijective, the orders of the two sets are equal.

$$ |\operatorname{cl}(a)| = |G/C(a)| = |G : C(a)| $$

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