# Order of Element divides Order of Centralizer

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## Theorem

Let $G$ be a finite group.

Let $x \in G$ be an element of $G$.

Let $\map {C_G} x$ denote the centralizer of $x$.

Then:

- $\order x \divides \order {\map {C_G} x}$

where:

- $\order x$ denotes the order of $x$ in $G$
- $\divides$ denotes divisibility
- $\order {\map {C_G} x}$ denotes the order of $\map {C_G} x$.

## Proof

From Order of Cyclic Group equals Order of Generator:

- $\order x = \order {\gen x}$

where $\gen x$ denotes the subgroup of $G$ generated by $x$.

By definition, $\gen x$ is a cyclic group.

By Cyclic Group is Abelian, all elements of $\gen x$ commute with $x$.

Thus by definition of centralizer:

- $\gen x \subseteq \map {C_G} x$

From Centralizer of Group Element is Subgroup, $\map {C_G} x$ is a subgroup of $G$.

So $\gen x$ is a subgroup of $\map {C_G} x$.

The result follows from Lagrange's theorem.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Cosets and Lagrange's Theorem: Exercise $14$