# Trivial group

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## Definition

### Verbal definition

The **trivial group** is the group with only one element, which is its identity element. The trivial group is usually denoted as , , or .

### 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
- The general linear group
- The orthogonal group of order 1 over a field of characteristic two

### Multiplication table

Element | (identity element) |
---|---|

## 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**.

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

order | 1 | only the identity element. |

exponent | 1 | |

nilpotency class | 0 | |

derived length | 0 | |

Frattini length | 0 | |

Fitting length | 0 | |

minimum size of generating set | 0 | |

subgroup rank of a group | 0 |

## Group properties

Property | Satisfied | Explanation |
---|---|---|

cyclic group | Yes | |

abelian group | Yes | |

nilpotent group | Yes | |

solvable group | Yes | |

perfect group | Yes |

## GAP implementation

### Group ID

This finite group has order 1 and has ID 1 among the groups of order 1 in GAP's SmallGroup library. For context, there are groups of order 1. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(1,1)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(1,1);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [1,1]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

The group can be defined using the TrivialGroup function:

`TrivialGroup`