In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that {\displaystyle b=g^{-1}ag.} This is an equivalence relation whose equivalence classes are called conjugacy classes.

What is conjugacy class of a group?

A conjugacy class of a group is a set of elements that are connected by an operation called conjugation. This operation is defined in the following way: in a group G, the elements a and b are conjugates of each other if there is another element g ∈ G g\in G g∈G such that a = g b g − 1 a=gbg^{-1} a=gbg−1.

How do you find the conjugacy class?

Conjugacy classes: definition and examples For an element g of a group G, its conjugacy class is the set of elements conjugate to it: {xgx-1 : x ∈ G}. Example 2.1. If G is abelian then every element is its own conjugacy class: xgx-1 = g for all x ∈ G.