Centre of a group

The centre $Z (𝐺)$ of a group $𝐺$ is a Normal subgroup of all elements of $𝐺$ that commute with every other element, i.e. $Z (𝐺) = {𝑎 \in 𝐺 ∣ 𝑎 𝑥 = 𝑥 𝑎 \forall 𝑥 \in 𝐺}$ . Note that at the very least $𝑒 \in Z (𝐺)$ .¹ group group

Proof of normal subgroup

As shown above, $𝑒 \in 𝑍 (𝐺)$ . Additionally, for any $𝑎, 𝑏 \in 𝑍 (𝐺)$ it is clear that $𝑎 𝑏 \in 𝑍 (𝐺)$ since $𝑎 𝑏 𝑥 = 𝑏 𝑥 𝑎 = 𝑥 𝑎 𝑏$ , so $𝑍 (𝐺)$ is closed under the binary operation. Since $𝑎 𝑥 = 𝑥 𝑎$ for any $𝑥 \in 𝐺$ we can both pre- and postmultiply both sides to obtain $𝑥 𝑎 - 1 = 𝑎 - 1 𝑥$ for any $𝑥$ , therefore $𝑎 - 1 \in 𝑍 (𝐺)$ , so $𝑍 (𝐺)$ is closed under the inverse. Hence $𝑍 (𝐺)$ is a subgroup of $𝐺$ by Two step subgroup test. Now let $𝑔 \in 𝐺$ and $ℎ \in 𝑍 (𝐺)$ . Clearly $𝑔 ℎ 𝑔 - 1 = 𝑔 𝑔 - 1 ℎ = ℎ \in 𝑍 (𝐺)$ . Hence $𝑍 (𝐺)$ is a Normal subgroup.

A related notion is the Centralizer in a group. The centre is the intersection of all centralisers.

Properties

The centre is necessarily abelian.

tidy | en | SemBr

2017, Contemporary Abstract Algebra, pp. 67 ↩

Table of Contents

Centre of a group

Properties

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Table of Contents

Centre of a group

Properties

Footnotes

Graph View

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