Centre of a group

Definition

Let $G$ be a group. The centre of $G$ is the set $Z(G) = \{z \in G : zg = gz \text{ for all } g \in G\}$.

Remarks

The centre is an abelian normal subgroup of $G$, and it is characteristic. The group $G$ is abelian if and only if $Z(G) = G$. The centre of the symmetric group $S_n$ is trivial for $n \geq 3$. The centre of $\mathrm{GL}_n(\mathbb{R})$ is the set of the nonzero scalar matrices $\lambda I$.

Explanations

L1 human

The centre collects the elements that commute with everybody. The identity is always in the centre. In an abelian group the centre is everything, and in many nonabelian groups it is only the identity.