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 $\{e\}$.

Alternative definitions

• The cyclic group on one element
• The symmetric group on one element
• The alternating group on one or two elements
• 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 $GL(1,2)$
• The orthogonal group of order 1 over a field of characteristic two

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

Linear representation theory

Further information: Linear representation theory of trivial group

The trivial group only has one irreducible representation, the trivial representation.

In Galois theory

Galois extensions with trivial Galois group

Further information: Galois extensions for trivial group

Given a Galois extension of fields $L/K$, the Galois group for the extension is the trivial group if and only if $L=K$.

Other results

The derived subgroup of a group is the trivial group precisely if that group is abelian, see the proof here.

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