Two members of a group, x and y, are said to be "conjugate" if there exists a member of the group, z, such that . That, of course, is the same as saying that . It is easy to prove that that is an equivalence relation (refexive: x is conjugate to itself by taking z equal to the group identity. symmetric: if x is conjugate to y, so that , then with . transitive: if x is conjugate to y and y is conjugate to z then and . Then so x is conjugate to x).
A "conjugacy class" is the equivalence class using "conjugate" as the equivalence relation.
Notice that if the group is Abelian (commutative) then zx= xz so that if x and y are conjuate, zx= xz= yz and then x= y. That, is, in Abelian groups, a member of the group is conjugate only to itself and the conjugacy classes consist only of the single elements.