Centralizer and normalizer

The centralizer is a term from the mathematical branch of group theory. The centralizer of an element of a group consisting of all existing with commuting group elements Quantity:

More generally defined as the centralizer of a subset of a group, the amount

Or equivalently the intersection of the centralizers of the individual elements of

Properties

The centralizer of a group element or a subset forms a subgroup of the group. In particular, the neutral element of a group in the centralizers is included each group member, and each subset, since it commutes with all group elements. The centralizer of the neutral element of a group, the group itself

For all elements of a group. For all elements of a group and all natural numbers and. Thus, for each element of a cyclic group generated by subgroup and subgroup of the centralizer of. For all elements of a subgroup of a group.

Conjugation

Each group acts on itself by conjugation. The centralizer of an element is then just the stabilizer with respect to this group action, ie the centralizer of an element of a group is the set of all group elements that leave unchanged under conjugation:

It follows that the number of elements that are to be conjugated, that is, the thickness of conjugacy of equal to the index of the centralizer is. In the case of a finite group which is the number of conjugated elements is always a divisor of the group order. If in a finite group is a set of representatives of all conjugacy classes of, then:

Center

For an abelian group, the group centralizers of all elements and all subsets are equal to the whole group. Conversely, the centralizer of any group (the set of commuting with all group elements group elements ) is always an abelian normal subgroup of the group. The centralizer is referred to as the center of the group. A group is abelian if and only if it is equal to its center.

A group element if and only contained in the center of the group, if its centralizer is equal to the whole group. The centralizer of a subset of a group is the largest subgroup of the group in which the elements are in the center of this subgroup.

Normalizer

Closely related to the concept of the centralizer is the notion of normalizer. In this case, the group on the set of its subgroups operated by conjugation. The centralizer is a normal subgroup in each normalizer.