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