# Conjugacy in a Group

__Conjugate Element__**:** If , then is said to be a conjugate of in if there exists an element such that .

Symbolically, we shall write for this and shall refer to this relation as conjugacy.

Then for some

__Theorem__**:** Conjugacy is an equivalence relation in a group.

__Proof__**:**

**(i) Reflexivity: **Let , then , hence , i.e. the relation of conjugacy is reflexive.

**(ii) Symmetric:** Let so that there exists an element such that . Now

Thus . Hence the relation is symmetric.

**(iii) Transitivity: **Let there exist two elements such that and for . Hence ,

Here and are the group. Therefore.

Hence the relation is transitive.

Thus conjugacy is an equivalence relation on .

**Conjugate Classes:** For , let , , the equivalence class of in under a conjugacy relation is usually called the conjugate class of in . It consists of the set of all distinct elements of the type .