Centralizer

The centralizer is a term from the mathematical branch of group theory . The centralizer of an element of a group is the set consisting of all group elements with commuting : ${\displaystyle Z_{G}(x)}$ ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle x}$

${\displaystyle Z_{G}(x):=\{g\in G\mid gx=xg\}}$

More generally, the centralizer of a subset of a group is defined as the set ${\displaystyle Z_{G}(U)}$ ${\displaystyle U}$ ${\displaystyle G}$

${\displaystyle Z_{G}(U):=\{g\in G\mid gu=ug\ \mathrm {f{\ddot {u}}r\ alle} \ u\in U\}}$

or, equivalently, the intersection of the centralizers of the individual elements ${\displaystyle U}$

${\displaystyle Z_{G}(U):=\bigcap _{x\in U}Z_{G}(x).}$

properties

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

The following applies to all elements of a group . For all elements of a group and all natural numbers we have and . Thus, for each element of a group, the cyclic subgroup generated by is also a subgroup of the centralizer of . The following applies to all elements of a subgroup of a group . ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle Z_{G}(x^{-1})=Z_{G}(x)}$ ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle n}$ ${\displaystyle x^{n}\in Z_{G}(x)}$ ${\displaystyle (x^{-1})^{n}\in Z_{G}(x)}$ ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle x}$ ${\displaystyle \langle x\rangle }$ ${\displaystyle Z_{G}(x)}$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle Z_{H}(x)=Z_{G}(x)\cap H}$

conjugation

→ Main article : Conjugation (group theory)

Each group operates on itself through conjugation. The centralizer of an element is then precisely the stabilizer with regard to this group operation, i. In other words, the centralizer of an element of a group is the set of all group elements that leave unchanged under conjugation: ${\displaystyle Z_{G}(x)}$ ${\displaystyle x}$ ${\displaystyle G}$ ${\displaystyle x}$

${\displaystyle Z_{G}(x)=\{g\in G\mid x=gxg^{-1}\}}$.

It follows that the number of elements to be conjugated, i.e. the cardinality of the conjugation class of , is equal to the index of the centralizer of . In the case of a finite group, the number of these conjugate elements is always a factor in the group order . If for a finite group there is a system of representatives of all conjugation classes of , then: ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle [G:Z_{G}(x)]}$ ${\displaystyle x}$ ${\displaystyle |G|}$ ${\displaystyle x_{1},x_{2},\dotsc ,x_{n}}$ ${\displaystyle G}$

${\displaystyle |G|=\sum _{j=1}^{n}[G:Z_{G}(x_{j})]}$

center

For an Abelian group , the centralizers of all group elements and all subsets are equal to the whole group . Conversely, the centralizer of any group (the set of group elements commuting with all group elements) is always an Abelian normal divisor of the group. The centralizer is called the center of the group. A group is Abelian if and only if it is equal to its center. ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle Z_{G}(G)}$ ${\displaystyle G}$ ${\displaystyle Z_{G}(G)}$ ${\displaystyle Z(G)}$ ${\displaystyle G}$

A group element is contained in the center of the group if and only 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 at the center of that subgroup.

Normalizer

Closely related to the concept of the centralizer is the concept of the normalizer . In this case the group operates on the set of its subgroups by conjugation . The centralizer is a normal divider in the respective normalizer.

literature

• Thomas W. Hungerford: Algebra . 5th edition. Springer, 1989, ISBN 0-387-90518-9 .

Individual evidence

1. ↑ Hungerford (1989), p. 89 f.