Let a be an element of a group G. The conjugacy class of a consists of all conjugates of a
Conj(a)={gag-1 such that g in G}

The conjugacy classes of the elements of G are equivalence classes for the equivalence relation on G given by aRb iff there exists x in G such that b=xax-1. As such the conjugacy classes of G partition G. See also centralizer.