Centralizer of Group Element is Subgroup

Theorem

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

Then $\map {C_G} a$, the centralizer of $a$ in $G$, is a subgroup of $G$.

Proof 1

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

We have that:

$\forall a \in G: e \circ a = a \circ e \implies e \in C_G \paren a$

Thus $C_G \paren a \ne \O$.

Let $x, y \in C_G \paren a$.

Then:

$\ds x \circ a$

$=$

$\ds a \circ x$

$\ds y \circ a$

$=$

$\ds a \circ y$

$\ds \leadsto \ \ $

$\ds x \circ y \circ a$

$=$

$\ds x \circ a \circ y$

$\ds $

$=$

$\ds a \circ x \circ y$

$\ds \leadsto \ \ $

$\ds x \circ y$

$\in$

$\ds C_G \paren a$

Thus $C_G \paren a$ is closed under $\circ$.

Let $x \in C_G \paren a$.

Then:

$\ds x \circ a$

$=$

$\ds a \circ x$

$\ds \leadsto \ \ $

$\ds x^{-1} \circ x \circ a \circ x^{-1}$

$=$

$\ds x^{-1} \circ a \circ x \circ x^{-1}$

$\ds \leadsto \ \ $

$\ds a \circ x^{-1}$

$=$

$\ds x^{-1} \circ a$

So:

$x \in C_G \paren a \implies x^{-1} \in C_G \paren a$

Thus, by the Two-Step Subgroup Test, the result follows.

$\blacksquare$

Proof 2

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

We have that:

$\forall a \in G: e \circ a = a \circ e \implies e \in \map {C_G} a$

Thus $\map {C_G} a \ne \O$.

Let $x, y \in \map {C_G} a$.

Then from Commutation with Group Elements implies Commuation with Product with Inverse:

$a \circ x \circ y^{-1} = x \circ y^{-1} \circ a$

so:

$x \circ y^{-1} \in\map {C_G} a$

The result follows by the One-Step Subgroup Test, the result follows.

$\blacksquare$

Sources

1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 8$: Compositions Induced on Subsets: Exercise $8.9$
1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.13$
1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $10$: The Orbit-Stabiliser Theorem: Exercise $4$
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): centralizer