Trivial group

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 on one element
 * The on one element
 * The of order 1 over any field
 * The of order 1 over any field
 * The $$GL(1,2)$$
 * The of order 1 over a field of characteristic two

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.

Other descriptions
The group can be defined using the TrivialGroup function:

TrivialGroup