DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11

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The conjugacy class $\operatorname{Cl}_G(x)$ and centralizer $C_G(x)$ of an element

Definition

Let $G$ be a group and $x\in G$ (Group and abelian group). The conjugacy class of $x$ is

$\operatorname{Cl}_G(x):=\{gxg^{-1}:g\in G\}.$

The centralizer of $x$ is

$C_G(x):=\{g\in G:gx=xg\}=\{g\in G:gxg^{-1}=x\}.$

The subgroup property implicit in the notation is proved in $C_G(x)$ and $N_G(H)$ are subgroups of $G$ ↗.

Depends on

Group and abelian group

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

T. W. Judson, Abstract Algebra: Theory and Applications, 14.2 (standard reference, not scraped)

The conjugacy class Cl⁡G(x)\operatorname{Cl}_G(x)ClG​(x) and centralizer CG(x)C_G(x)CG​(x) of an element