# Trivial group

It has been suggested that this article be merged with trivial ring and trivial module to Zero object (algebra). (Discuss) Proposed since February 2012. |

In mathematics, a **trivial group** is a group consisting of a single element. All such groups are isomorphic, so one often speaks of *the* trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: 0, 1 or *e* depending on the context. If the group operation is denoted ∗ then it is defined by *e* ∗ *e* = *e*.

The similarly defined **trivial monoid** is also a group since its only element is its own inverse, and is hence the same as the trivial group.

The trivial group should not be confused with the empty set (which has no elements, and lacking an identity element, cannot be a group).

Given any group *G*, the group consisting of only the identity element is a subgroup of *G*, and, being the trivial group, is called the **trivial subgroup** of *G*.

The term, when referred to "*G* has no nontrivial proper subgroups" refers to the fact that all subgroups of *G* are the trivial group {*e*} and the group *G* itself.

## Properties

The trivial group is cyclic of order 1; as such it may be denoted Z_{1} or C_{1}. If the group operation is called addition, the trivial group is usually denoted by 0. If the group operation is called multiplication then 1 can be a notation for the trivial group.

The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.

## See also

- List of small groups

## References

- Rowland, Todd and Weisstein, Eric W., "Trivial Group",
*MathWorld*.