# Trivial group

From Groupprops

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this groupView a complete list of particular groups (this is a very huge list!)[SHOW MORE]

## Definition

The **trivial group** is the group with only one element, which is its identity element.

### Alternative definitions

- The cyclic group on one element
- The symmetric group on one element
- The projective general linear group of order 1 over any field
- The special linear group of order 1 over any field

## Importance

The trivial group is important in the following ways:

- For any group, there is a unique homomorphism from the trivial group to that group, namely the homomorphism sending it to the identity element. Thus, the trivial group occurs in a unique way as a subgroup for any given group, namely the one-element subgroup comprising the identity element. This is termed the
**trivial subgroup**.

- For any group, there is a unique homomorphism to the trivial group from that group, namely the homomorphism sending everything to the identity element. Thus, the trivial group occurs in a unique way as a quotient group of any given group, namely its quotient by itself. This is termed the
**trivial quotient**.