---
tag: 0S7Y9Q990Q
kind: definition
name: Centre of a group
msc: [20A05]
defines: 0DZ93493VN
uses: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Center_(group_theory)
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Center_(group_theory)
    locator: As a subgroup
---

## Definition

Let $G$ be a [[group|G5S1M9]].
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|082GHNPJ53]] [[normal subgroup|0CB0DX5BVV]] of $G$, and it is [[characteristic|0KABPMRHM2]].
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.