# 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 $1$, $\{ 1 \}$, or $\{ e \}$.

### Multiplication table

Element $e$ (identity element) $e$ $e$

## 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 $G$:

gap> G := SmallGroup(1,1);

Conversely, to check whether a given group $G$ 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