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.
(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 .