Trivial group
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Contents
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 | ![]() |
---|---|
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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