Order of Element divides Order of Centralizer

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 $14$