Dogmathic

In group theory, the centralizer of an element tells us which elements of the group commute with it. For an element $a$ in a group $G$, the centralizer $C_G(a)$ is the set of all $x$ in $G$ such that $xa = ax$. In this post, we’ll look at what that definition means, work through examples, and show why the centralizer is always a subgroup of $G$.

Definition of the Centralizer

For an element $a$ in a group $G$, the centralizer of $a$ is the set of all elements of $G$ that commute with $a$.

$C_G(a) = \{x ∈ G : xa = ax\}$

The important idea is that we fix one element, $a$, and then ask:

Which elements of $G$ can switch places with $a$ without changing the product?

Those elements form $C_G(a)$.

A Simple Example

Suppose $G$ is an abelian group. By definition, every pair of elements in $G$ commutes.

So for every $a ∈ G$ and every $x ∈ G$,

$xa = ax$.

That means every element of $G$ belongs to the centralizer of $a$.

Therefore:

$C_G(a) = G$

for every $a$ in an abelian group.

The centralizer becomes more interesting when $G$ is nonabelian, because some elements may commute with a while others do not.

Why Is the Centralizer a Subgroup?

The centralizer $C_G(a)$ is not just a collection of elements that happen to commute with $a$. It is always a subgroup of $G$.

Using the subgroup test, take$x, y ∈ C_G(a)$. Since both commute with$a$,

$xa = ax$

and

$ya = ay.$

From $ya = ay$, it follows that$y⁻¹$ also commutes with $a$.

Now consider $xy⁻¹$:

$(xy⁻¹)a = x(y⁻¹a) = x(ay⁻¹) = (xa)y⁻¹ = (ax)y⁻¹ = a(xy⁻¹).$

Therefore $xy⁻¹$ also commutes with $a$, so

$xy⁻¹ ∈ C_G(a)$.

By the subgroup test,

$C_G(a) ≤ G.$

Centralizer vs. Center of a Group

The centralizer and the center are closely related, but they are not the same thing.

The centralizer $C_G(a)$ contains the elements that commute with one particular element $a$.

The center $Z(G)$ contains the elements that commute with every element of $G$.

So the center can be written as the intersection of all the centralizers:

$Z(G) = ⋂ C_G(a)$

where the intersection is taken over all $a ∈ G$.

This gives another way to think about the center: it consists of the elements that survive every possible centralizer condition.

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