Conjugation (group theory)

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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

or as a legal operation

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.

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:

  • Each element is conjugated to itself ( reflexivity ).
  • Is conjugated to , is also conjugated to ( symmetry ).
  • Is conjugated to and conjugated to , then is also conjugated to ( transitivity ).

All elements that are conjugated to one another each form an equivalence class , the so-called conjugation class of :

In this case, as the conjugacy be selected an arbitrary element. The conjugation classes are the lanes of the conjugation operation.

The stabilizer

of an element is the centralizer of .

Two subgroups and a group are called conjugate to each other if there is one with .

A subgroup of a group is invariant under conjugation if for all elements from and all elements from the product is again in :

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 .


The conjugation with is the picture


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

one gets the center of . According to the homomorphism theorem , the mapping thus conveys an isomorphism from to .

Individual evidence

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