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.

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