trivial group

- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space

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.

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.

The trivial group is an example of a trivial algebra.

Last revised on July 5, 2021 at 15:11:01. See the history of this page for a list of all contributions to it.