# Conjugation (group theory)

The conjugation operation is a group operation that divides a group into conjugation classes . The elements of a conjugation class have a lot in common, so that a closer look at these classes provides important insights into the structure of non-Abelian groups. Conjugation classes are irrelevant for Abelian groups , as each group element forms its own conjugation class.

## Conjugation operation

The conjugation operation is an operation of a group on itself, either as a left operation

${\ displaystyle (g, h) \ mapsto ghg ^ {- 1}}$

or as a legal operation

${\ displaystyle (g, h) \ mapsto h ^ {- 1} gh}$

is defined.

The exponential notation is common for the right operation . In this notation the conjugation operation satisfies the relationship . In the following, the conjugation operation is defined as a left operation. ${\ displaystyle (g, h) \ mapsto h ^ {- 1} gh}$${\ displaystyle h ^ {- 1} gh = g ^ {h}}$${\ displaystyle (x ^ {g}) ^ {h} = x ^ {gh}}$

Two elements and a group G are said to be conjugated to one another if there is an element such that is. The conjugation is an equivalence relation . So it has the following properties: ${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$${\ displaystyle g \ in G}$${\ displaystyle h_ {1} = gh_ {2} g ^ {- 1}}$

• Each element is conjugated to itself ( reflexivity ).${\ displaystyle h}$
• Is conjugated to , is also conjugated to ( symmetry ).${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$${\ displaystyle h_ {2}}$${\ displaystyle h_ {1}}$
• Is conjugated to and conjugated to , then is also conjugated to ( transitivity ).${\ displaystyle h_ {1}}$${\ displaystyle h_ {2}}$${\ displaystyle h_ {2}}$${\ displaystyle h_ {3}}$${\ displaystyle h_ {1}}$${\ displaystyle h_ {3}}$

All elements that are conjugated to one another each form an equivalence class , the so-called conjugation class of : ${\ displaystyle h}$

${\ displaystyle G \ cdot h = \ left \ {ghg ^ {- 1} \ mid g \ in G \ right \}}$

In this case, as the conjugacy be selected an arbitrary element. The conjugation classes are the lanes of the conjugation operation. ${\ displaystyle h}$

The stabilizer

${\ displaystyle Z_ {G} (x) = \ left \ {g \ in G \ mid x = gxg ^ {- 1} \ right \}}$

of an element is the centralizer of . ${\ displaystyle x}$${\ displaystyle x}$

Two subgroups and a group are called conjugate to each other if there is one with . ${\ displaystyle U}$${\ displaystyle V}$${\ displaystyle G}$${\ displaystyle g \ in G}$${\ displaystyle V = gUg ^ {- 1}}$

A subgroup of a group is invariant under conjugation if for all elements from and all elements from the product is again in : ${\ displaystyle N}$${\ displaystyle G}$${\ displaystyle h}$${\ displaystyle N}$${\ displaystyle g}$${\ displaystyle G}$${\ displaystyle ghg ^ {- 1}}$${\ displaystyle N}$

${\ displaystyle gNg ^ {- 1} = N}$

A subgroup of a group that is invariant under conjugation is called the normal subgroup of the group. Normal divisors allow the group to form factor groups .

## conjugation

The conjugation with is the picture ${\ displaystyle g}$

${\ displaystyle \ operatorname {int} _ {g} \ colon G \ rightarrow G, \ quad h \ mapsto ghg ^ {- 1}}$.

It arises from the conjugation operation by holding . The conjugation is an automorphism of . Automorphisms of , which can be written as a conjugation with an element of , are called inner automorphisms . This is where the name comes from , where the "int" stands for "interior". The inner automorphisms form a normal divisor of the automorphism group of . As the core of group homomorphism${\ displaystyle g}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle G}$${\ displaystyle \ operatorname {int} _ {g}}$${\ displaystyle \ operatorname {Inn} (G)}$${\ displaystyle G}$

${\ displaystyle T \ colon \ G \ rightarrow \ operatorname {Inn} (G), \ quad g \ mapsto \ operatorname {int} _ {g}}$

one gets the center of . According to the homomorphism theorem , the mapping thus conveys an isomorphism from to . ${\ displaystyle Z (G)}$${\ displaystyle G}$${\ displaystyle T}$${\ displaystyle G / Z (G)}$${\ displaystyle \ operatorname {Inn} (G)}$

## Individual evidence

1. ^ Siegfried Bosch : Algebra . Springer, 2004, ISBN 3-540-40388-4 , p. 239