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.

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^-1yg=x

⇒ g^-1y(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^-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.

∴ 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^-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.

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/C_G(x), where C_G(x) is the centralizer of x in G.. In other words, |Cl(x)| = [G : C_G(x)].
5. 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

Basics of Group Theory

Abelian Group

Cyclic Group

Group Homomorphism

First Isomorphism Theorem of Groups

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