Conjugacy class

The Konjugationsoperation is a group operation, which breaks a group into conjugacy classes. The elements of a conjugacy class have a lot in common, so that a closer examination of these classes allows important insights into the structure of non- abelian groups. In Abelian groups, conjugacy classes are irrelevant since each group element forms its own conjugacy class.

Konjugationsoperation

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

Or as a right action

Is defined.

For the right operation, exponential notation is common. In this notation, Konjugationsoperation satisfies the relationship. The following are the Konjugationsoperation is defined as a left operation.

Two elements and a group called conjugate to each other if there is an element, so is. The conjugacy is an equivalence relation. Therefore it has the following properties:

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

All elements that are conjugate to each other, each equivalence class, called the conjugacy class of form:

It can be chosen as any element of the conjugacy class. The conjugacy classes are the orbits of the Konjugationsoperation.

The stabilizer

Is an element of the centralizer of.

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

A invariant under conjugation subgroup of a group is called a normal subgroup of the group. Normal subgroup allow the formation of factor groups of the group.

Conjugation

The conjugation with the mapping

It arises from the Konjugationsoperation by detained. The conjugation is a so-called inner automorphism of. Comes the term, wherein the "int" for " interior" is.

The figure

Is in the group of inner automorphisms, a normal subgroup of the automorphism from.

The core of the center of:

After the homomorphism the picture conveys so after an isomorphism of.

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