Definition:Center (Abstract Algebra)/Group

''This page is about the centers of Groups and Rings. For the center of a Circle, see Center of a Circle.''

Center of a Group
The center of a group $$G$$, denoted $$Z \left({G}\right)$$, is the subset of elements in $$G$$ that commute with every element in $$G$$. Symbolically:


 * $$Z \left({G}\right) = C_G \left({G}\right) = \left\{{g \in G: g x = x g, \forall x \in G}\right\}$$.

That is, the center of $$G$$ is the centralizer of $$G$$ in $$G$$ itself.

It is shown here that $$Z \left({G}\right) \triangleleft G$$ for any group $$G$$.

Center of a Ring
The center of a ring $$\left({R, +, \circ}\right)$$, denoted $$Z \left({R}\right)$$, is the subset of elements in $$R$$ that commute with every element in $$R$$. Symbolically:


 * $$Z \left({R}\right) = C_R \left({R}\right) = \left\{{x \in R: \forall s \in R: s \circ x = x \circ s}\right\}$$

That is, the center of $$R$$ is the centralizer of $$R$$ itself.