# Trivial group

In **group **theory , the **trivial group** is a group whose carrier set contains exactly one element . The trivial group is clearly defined except for isomorphism . Each group contains the trivial group as a subgroup .

## definition

The trivial group is a group that consists of the one-element set and is provided with the only possible group operation

- .

The element is thus the neutral element of the group.

## Examples

All trivial groups are isomorphic to one another . Examples of trivial groups are:

- the cyclic group of the degree
- the alternating group of degree
- the symmetric group of a one-element set

## properties

- Since the group operation is commutative, the trivial group is an Abelian group .
- The only subgroup of the trivial group is the trivial group itself.
- The trivial group of the empty set generated : . Here, according to the usual convention , the empty product results in the neutral element.
- Each group contains the trivial group and itself as the ( trivial ) normal divisor . The trivial group is therefore mostly not seen as a simple group .
- In the category of groups
**Grp**, the trivial group functions as a null object .

## See also

## literature

- Rainer Schulze-Pillot: Introduction to Algebra and Number Theory . Springer, 2008, ISBN 3-540-79570-7 .
- Jürgen Wolfart: Introduction to number theory and algebra . Springer, 2010, ISBN 3-8348-9833-3 .

## Web links

- Todd Rowland, Eric W. Weisstein :
*Trivial Group*. In:*MathWorld*(English).