nLab trivial group

Definition

The trivial group is the group whose underlying set is the singleton, hence whose only element is the neutral element.

In the context of nonabelian groups the trivial group is usually denoted $1$, while in the context of abelian groups it is usually denoted $0$ (being the zero object) and also called the zero group (notably in homological algebra).

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

Examples

The trivial group is a subgroup of any other group, and the corresponding inclusion $1 \hookrightarrow G$ is the unique such group homomorpism.

The quotient group of any group $G$ by itself is the trivial group: $G/G = 1$, and the quotient projection $G \to G/G =1$ is the unique such group homomorphism.

It can be nontrivial to decide from a group presentation whether a group so presented is trivial, and in fact the general problem is undecidable. See also combinatorial group theory and word problem.

Properties

The trivial group is an example of a trivial algebra.

Related concepts

• Andrews-Curtis conjecture

• trivial monoid

• terminal category