# 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${\ displaystyle (\ {e \}, *)}$ ${\ displaystyle \ {e \}}$

${\ displaystyle e * e = e}$.

The element is thus the neutral element of the group. ${\ displaystyle e}$

## Examples

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

• the cyclic group of the degree${\ displaystyle C_ {1}}$${\ displaystyle 1}$
• the alternating group of degree${\ displaystyle A_ {2}}$${\ displaystyle 2}$
• the symmetric group of a one-element set${\ displaystyle S_ {1}}$

## properties

• Since the group operation is commutative, the trivial group is an Abelian group .${\ displaystyle \ ast}$
• 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.${\ displaystyle \ {e \} = \ langle \ emptyset \ rangle}$
• 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 .