nLab
trivial group

The trivial group is the point $•$ interpreted as a group, often denoted $1$ or $0$. Its underlying set is a singleton, and its unique element is the identity.

The trivial group is a zero object (both initial and terminal) of Grp.

Given any group $G$, the unique group homomorphisms from $1$ to $G$ and from $G$ to $1$ make $1$ both a subgroup and a quotient group of $G$. In such a guise, it is called the trivial subgroup or trivial quotient group of $G$; the former is also called the identity subgroup.

We also denote the trivial group as $\{1\}$ or $\{0\}$, especially when viewed as a trivial subgroup. The trivial quotient group of $G$ may be denoted $G/G$ or $\{G\}$.

The trivial group is an example of a trivial algebra.