Conjugacy classes

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Section 1.7 Conjugacy classes

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Objectives

You should be able to:

Determine the conjugacy classes of a finite group.

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Subsection 1.7.1 Definition

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We have already discussed briefly the idea of equivalence classes. Given a group

$G\text{,}$ the idea is to define an equivalence relation, which we generally denote by

$\equiv\text{,}$ between elements of the group, which is reflexive (

$a \equiv a$ ), symmetric (if

$a \equiv b$ then

$b \equiv a$ ) and transitive (if

$a \equiv b$ and

$b \equiv c$ then

$a \equiv c$ ). Once we have an equivalence relation, we can define equivalence classes, which are subsets of elements of the group that are equivalent according to

$\equiv\text{.}$

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There is a particularly interesting equivalence relation, known as conjugation, that partitions any finite group into equivalence classes, known as conjugacy classes.

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Definition 1.7.1. Conjugacy classes.

We say that two elements $a,b \in G$ are conjugate if there exists an element $g \in G$ such that $a = g b g^{-1}\text{.}$ This is an example of an equivalence relation. Then we can form equivalence classes, that is, classes of elements of $G$ that are conjugate to each other. The set of all elements conjugate to one another is called a conjugacy class.

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Conjugacy classes are composed of elements that somehow behave in a similar way. What “similar” means here depends on the context. As we will see, for the symmetric group

$S_n\text{,}$ conjugacy classes will be comprised of permutations that have the same cycle structure. In the group of three-dimensional rotations

$SO(3)\text{,}$ we can characterize rotations by specifying an axis of rotation and an angle of rotation. Then one can show that all rotations that have the same angle belong to the same conjugacy class.

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As we will see, conjugacy classes are very useful in the study of groups, especially in representation theory.

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Subsection 1.7.2 Properties of conjugacy classes

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It is clear that conjugacy classes are disjoint, since an element cannot be in two different conjugacy classes. So we can partition a group

$G$ into its conjugacy classes.

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It is also interesting to note that all elements in a given conjugacy class have the same order.

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Definition 1.7.2. Order of an element of a group.

We define the order of an element $a \in G$ to be the smallest integer $n$ such that $a^n = e\text{.}$

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With this definition we can prove the following lemma:

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Lemma 1.7.3.

All elements of a given conjugacy class have the same order.

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Proof.

Suppose that $a \in G$ has order $n\text{.}$ If $b$ is conjugate to $a\text{,}$ that is $b = g a g^{-1}$ for some $g \in G\text{,}$ then

\begin{equation*} b^n = (g a g^{-1} )^n = (g a g^{-1}) (g a g^{-1}) \cdots (g a g^{-1}) = g a^n g^{-1} = g e g^{-1} = e. \end{equation*}

It follows that $b^n = e\text{,}$ and hence the order of $b$ is $\leq n\text{.}$

To finish the proof, we need to argue that there cannot be an integer $m \lt n$ such that $b^m= e\text{,}$ so that the order of $b$ is precisely $n\text{.}$ Suppose that there is such a $m \lt n\text{.}$ Then

\begin{equation*} b^m = g a^m g^{-1} = e. \end{equation*}

Multiplying by $g^{-1}$ on the left and $g$ on the right, we get that $a^m = e\text{.}$ But $n$ must be the smallest integer such that $a^n = e\text{,}$ therefore this is a contradiction. It follows that the order of $b$ is equal to $n\text{.}$

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We can also prove the following interesting result:

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Lemma 1.7.4.

Each element in the centre of a group $G$ constitutes a conjugacy class by itself. And hence the identity is always a conjugacy class by itself, and for an Abelian group $G\text{,}$ each element forms its own distinct class.

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Proof.

Let $h \in Z(G)$ be an element of the centre of $G\text{.}$ Then for all $g \in G\text{,}$ we have $h g = g h\text{,}$ and hence for all $g \in G\text{,}$ $g h g^{-1} = h\text{,}$ therefore the only element conjugate to $h$ is $h$ itself.

The two corollaries follow directly, since the identity is always an element of the centre of a group, and if $G$ is abelian, then $Z(G) = G\text{.}$