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From Groupprops

This article describes an equivalence relation on the set of elements of a group

Contents 1 Definition 1.1 Symbol-free definition 1.2 Definition with symbols 1.3 Equivalence of definitions

Definition

Symbol-free definition

Given a group, two (possibly equal) elements of the group are termed conjugate elements if the following equivalent conditions are satisfied:

1. There is an inner automorphism of the group mapping one element to the other
2. There are two elements of the group whose products, in the two possible orders, give these two elements

Definition with symbols

Given a group G and elements , g is termed conjugate to h if the following equivalent conditions are satisfied:

1. There exists such that xgx ^{− 1} = h, in other words, the inner automorphism of conjugation by x, sends g to h
2. There exist such that g = ab,h = ba

The equivalence classes under the equivalence relation of being conjugate are termed the conjugacy classes.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of conjugate elements