Definition:Center (Abstract Algebra)/Group

This page is about the center of a Group.

For the center of a ring, see Center of a Ring.

For the center of a circle, see Center of a Circle.

Definition

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.

Also known as

Some sources use $Z_G$ to denote this concept.

Linguistic Note

The UK English spelling of this is centre.

Also see

• Center is a Normal Subgroup: $Z \left({G}\right) \triangleleft G$ for any group $G$.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): Exercise $12.11$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 1.9$: Exercise $5.14$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$: Problem $\text{AA}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 35 \delta$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 50$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 37 \ (2)$
• John F. Humphreys: A Course in Group Theory (1996): $\S 7$: Exercise $5$