This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This article is about a particular group, i.e., a group unique upto isomorphism.
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The trivial group is the group with only one element, which is its identity element. The trivial group is usually denoted as , , or .
- 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
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.
|order||1||only the identity element.|
|minimum size of generating set||0|
|subgroup rank of a group||0|
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:
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:
to have GAP output the group ID, that we can then compare to what we want.
The group can be defined using the TrivialGroup function: