For two elements x, y in a group G are said to be conjugate if gxg^{-1}=y for some g in G. This relation is the conjugacy relation in group theory. It defines an equivalence relation, so the group G can be decomposed into different conjugacy classes. This way we will obtain the class equation of G if the group is finite. In this article, we will learn about the conjugacy relation and classes together with the class equation of groups.

## Conjugacy Relation in Groups

An element y of a group G is said to be conjugate to another element x of G if there exists an element g in G such that

y = gxg^{-1}.

We define the above relation by ρ. That is,

xρy if and only if y = gxg^{-1}_{ }for some g ∈ G.

## Conjugacy is an Equivalence Relation

**Theorem: **The above conjugacy relation ρ on G is an equivalence relation.

*Proof:*

**Reflexive:** Note that x=exe^{-1} for any x ∈ G.

Hence x is conjugate to x, that is, xρx holds.

∴ ρ is reflexive.

**Symmetric:** Let x, y ∈ G be such that xρy holds, that is,

y = gxg^{-1}_{ }for some g ∈ G.

⇒ g^{-1}yg=x

⇒ g^{-1}y(g^{-1})^{-1}=x as we know that (g^{-1})^{-1}=g.

Thus, xρy holds ⇒ yρx holds.

∴ ρ is symmetric.

**Transitive:** Let x, y, z ∈ G be such that xρy and yρz hold, that is,

y = gxg^{-1}_{ }and z=hyh^{-1} for some g,h ∈ G.

Now, z=hyh^{-1} =h(gxg^{-1})h^{-1} = (hg)x(hg)^{-1} as we know that (hg)^{-1}=g^{-1}h^{-1}.

Thus, z=(hg)x(hg)^{-1} for some hg ∈ G.

⇒ xρz holds.

Hence, we have shown that xρy and yρz hold ⇒ xρz holds.

∴ ρ is transitive.

Thus, we have proved that ρ is reflexive, symmetric, and transitive. So the conjugacy relation ρ is an equivalence relation.

## Conjugacy Class

The conjugacy relation ρ defined by xρy iff y = gxg^{-1}_{ }for some g ∈ G is an equivalence relation. Then the group G is partitioned into ρ-equivalence classes called the **conjugacy classes**.

For x ∈ G, the conjugacy class of x is denoted by Cl(x) and is defined as follows:

Cl(x) = {gxg^{-1}: g ∈ G}.

## Conjugacy Class Examples

Conjugacy classes of cyclic groups: Let G be a cyclic group. As cyclic groups are abelian, we have gh=hg ∀ g, h∈ G.

Let x∈ G be an element of G. Then the conjugacy class of x in G is given by

Cl(x) = {gxg^{-1}: g ∈ G}

= {gg^{-1}x: g ∈ G} as G is abelian.

= {x}.

So every conjugacy class in a cyclic group contains only one element. Thus, the number of conjugacy classes of a cyclic group of order n is equal to n.

For the above reason, every conjugate class of the centre Z(G):={x∈ G: xg=gx ∀ g∈ G} of a group contains only one element as Z(G) is abelian.

## Conjugacy Class Properties

- The conjugacy class of the identity element is the identity itself. That is, Cl(e)={e}.
- If G is an abelian group, then Cl(x)={x} for all x ∈ G.
- If an element of a group is conjugate to itself, then that element is called self-conjugate. As the conjugacy class Cl(x) for x∈ Z(G), the centre of G, contains x only, they are called self-conjugate elements.
- Let G be a finite group and x∈G. Then the cardinality of Cl(x) is equal to the cardinality of the quotient group G/C
_{G}(x), where C_{G}(x) is the centralizer of x in G.. In other words, |Cl(x)| = [G : C_{G}(x)]. - In a finite group, |Cl(x)| is always a divisor of |G|. Because |G| = |Cl(x)| × |C
_{G}(x)|.

## Class Equation of a Group

Let G be a group with centre Z(G). If x_{i} ∉ Z(G) are the representatives of the distinct conjugacy classes, then from the disjointness of the conjugacy classes the class equation of G is given by

|G| = |Z(G)| + ∑_{i} [G: C_{G}(x_{i})].

**Also Read**

First Isomorphism Theorem of Groups

## FAQs

**Q1: How do define conjugacy relation?**

Answer: Two elements x, y of a group G are called conjugate to each other if y = gxg^{-1}_{ }for some g ∈ G.

**Q2: What is the class equlation of S**

_{3}?Answer: The class equation of S_{3} is 6=|S_{3}|=1+1+2+2.