Definition:Center (Abstract Algebra)
This page is about centers in the context of Abstract Algebra. For other uses, see Center.
Definition
Semigroup
The center of a semigroup $\struct {S, \circ}$, denoted $\map Z S$, is the subset of elements in $S$ that commute with every element in $G$.
Symbolically:
- $\map Z S = \set {s \in S: \forall x \in S: s \circ x = x \circ s}$
Group
The center of a group $G$, denoted $\map Z G$, is the subset of elements in $G$ that commute with every element in $G$.
Symbolically:
- $\map Z G = \map {C_G} G = \set {g \in G: g x = x g, \forall x \in G}$
That is, the center of $G$ is the centralizer of $G$ in $G$ itself.
Ring
The center of a ring $\struct {R, +, \circ}$, denoted $\map Z R$, is the subset of elements in $R$ that commute under $\circ$ with every element in $R$.
Symbolically:
- $\map Z R = \map {C_R} R = \set {x \in R: \forall s \in R: s \circ x = x \circ s}$
Also see
- Definition:Centralizer
- Results about centers (in the context of abstract algebra) can be found here.
Historical Note
The notation $\map Z S$, conventionally used for the center of a structure $\struct {S, \circ}$, derives from the German Zentrum, meaning center.
Linguistic Note
The British English spelling of center is centre.
The convention on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to use the American English spelling center, but it is appreciated that there may be lapses.