Centralizer in a group

The centralizer $𝐶 𝐺 (𝑎)$ of an element $𝑎 \in 𝐺$ is a Subgroup of $𝐺$ containing all elements that commute with $𝑎$ ,¹ #m/def/group i.e.

𝐶 𝐺 (𝑎) = {𝑏 \in 𝐺 ∣ 𝑎 𝑏 = 𝑏 𝑎}

More generally, the centralizer $𝐶 𝐺 (𝑆)$ of any set $𝑆 \subseteq 𝐺$ contains elements which commute with the whole of $𝑆$ , i.e.

𝐶 𝐺 (𝑆) = {𝑥 \in 𝐺 : (\forall 𝑠 \in 𝑆) [𝑥 𝑠 = 𝑠 𝑥]}

Proof of subgroups

Let $𝑎 \in 𝐺$ . Clearly $𝑒 \in 𝐶 (𝑎)$ , Given any $𝑏, 𝑐 \in 𝐶 (𝑎)$ , clearly $(𝑏 𝑐) 𝑎 = 𝑏 (𝑐 𝑎) = 𝑏 (𝑎 𝑐) = (𝑏 𝑎) 𝑐 = (𝑎 𝑏) 𝑐 = 𝑎 (𝑏 𝑐)$ , hence $𝐶 (𝑎)$ is closed under the binary operation. Similarly, $𝑎 𝑏 = 𝑏 𝑎$ may be both pre- and postmultiplied by $𝑏 - 1$ to obtain $𝑏 - 1 𝑎 = 𝑎 𝑏 - 1$ , so $𝐶 (𝑎)$ is closed under the inverse operation. Hence $𝐶 (𝑎)$ is a subgroup of $𝐺$ by Two step subgroup test.

Since the intersection of subgroups is a subgroup, $𝐶 𝐺 (𝑆) = ⋂ 𝑠 \in 𝑆 𝐶 𝐺 (𝑠)$ must also be a subgroup.

A related notion is the Centre of a group $𝑍 (𝐺) = 𝐶 𝐺 (𝐺)$ , which includes only those elements that commute with all group elements.

Additional properties

Clearly the centraliser itself need not be abelian, since the centraliser of any $𝑧 \in 𝑍 (𝐺)$ will be the entire group. For example, in the non-abelian group $𝐷 4$ , the centraliser $𝐶 (𝑅 180 o̲) = 𝐷 4$ .

#state/tidy | #lang/en | #SemBr

2017, Contemporary Abstract Algebra, p. 68 ↩

Centralizer in a group

Additional properties

Footnotes