Trivial Group White Sheet [Printable Version]

Alternate Descriptions: (* Most common)

Name	Symbol(s)
* Trivial Group	* 0, 1
Cyclic 1	C1, Z1
Symmetric 0, 1	S0, S1
Dihedral 0	D0
Quaternion 0	Q0

Most authors prefer the use of 1 for the trivial element of a group; consequently 1 is a natural choice for the trivial group. However, the use of the symbol 0 has categorical implications. It reminds the reader that the object is a zero object in the category of all groups. This is useful when expressing short exact sequences and the like.

GAP ID: [1,1]
Magma ID:?

Standard Invariants:

Invariant	Value
Abelian	True
Nilpotent	True
Nilpotency Class	0
Ascending Central Series	[0]
Descending Central Series	[0]
Solvable	True
Composition Factors	{}
Automorphism Group	0

Permutation Representations [Description]

Action Name	Degree	Generators	Stabilizer (Isomorphism Type)	Stabilizer (Genertors)	Primitivity Blocks
	[Description]		[Description]	[Description]	[Description]
Trivial	0	()	0	()	{}
Regular	1	(1)	0	()	{1}

Note that the regular action of the trivial group is not primitive as it does not have precisely two block systems of imprimitivity, it has only 1. In the same way the trivil group is definitionally excluded from being a simple group.