Definition:Centralizer/Group Element

Definition

Let $\struct {G, \circ}$ be a group.

Let $a \in \struct {G, \circ}$.

The centralizer of $a$ (in $G$) is defined as:

$\map {C_G} a = \set {x \in G: x \circ a = a \circ x}$

That is, the centralizer of $a$ is the set of elements of $G$ which commute with $a$.

Also known as

Some sources call this the normalizer of $a$ in $G$ but that term generally has another meaning.

Also see

• Stabilizer of Element under Conjugacy Action is Centralizer

• Results about centralizers can be found here.

Linguistic Note

The UK English spelling of centralizer is centraliser.

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.6$. Stabilizers: Example $108$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Exercise $25.16 \ \text{(b)}$
• 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.13$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 37$. Some important general examples of subgroups $(1)$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Exercise $(11)$
• 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Example $10.10$