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