Conjugacy Relation, Class Equation of Groups with Examples

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.


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-1yg=x

⇒ g-1y(g-1)-1=x as we know that (g-1)-1=g. Thus, xρy holds ⇒ yρx holds. This prove thaρ 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-1h-1.

Thus, z=(hg)x(hg)-1 for some hg ∈ G.

⇒ xρz holds.

Hence, we have shown that xρy and yρz hold ⇒ xρz holds, that proves ρ is transitive.

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

Read This: Basics of Group Theory

Every Subgroup of a Cyclic Group is Cyclic: Proof

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}.

More Topic: Abelian Group | Cyclic Group

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-1x: 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.

Have You Read These?

Conjugacy Class Properties

  1. The conjugacy class of the identity element is the identity itself. That is, Cl(e)={e}.
  2. If G is an abelian group, then Cl(x)={x} for all x ∈ G.
  3. 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.
  4. 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/CG(x), where CG(x) is the centralizer of x in G.. In other words, |Cl(x)| = [G : CG(x)].
  5. In a finite group, |Cl(x)| is always a divisor of |G|. Because |G| = |Cl(x)| × |CG(x)|.

Also Read: Center of a Group | Normalizer of a Group

Class Equation of a Group

Let G be a group with centre Z(G). If xi ∉ 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: CG(xi)].

Also Read

Group Homomorphism

First Isomorphism Theorem of Groups


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 S3?

Answer: The class equation of S3 is 6=|S3|=1+1+2+2.

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