---
tag: 0BNPC9850Y
kind: example
name: Trivial group
aliases: [zero-group]
msc: [20A05]
instance_of: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Trivial_group
    locator: lead section
same_as:
  wikidata: Q568687
  wikipedia: Trivial group
---

## Example

The trivial group is the [[group|G5S1M9]] $\{e\}$ with one element $e$ and the operation $e \cdot e = e$.
The element $e$ is the identity element, and it is its own inverse.
Each group with exactly one element is isomorphic to this group.

## Remarks

Each group $G$ contains a trivial subgroup $\{e\}$, where $e$ is the identity of $G$.
In additive notation the trivial group is written $0$ or $\{0\}$.
The trivial group is [[abelian|082GHNPJ53]], and its order is $1$.

## Explanations

### L1 human

The trivial group is the smallest possible group: it has only the identity element.
It satisfies all the group axioms in a simple way.
It is often an edge case in theorems, for example when a theorem asks for a group that is not trivial.