Trivial group

Example

The trivial group is the group $\{e\}$ 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, 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.