Trivial group

The trivial group is in group theory, a group whose underlying set contains exactly one element. The trivial group is uniquely determined up to isomorphism. Each group contains the trivial group as a subgroup.

Definition

The trivial group is a group consisting of the singleton provided with the only possible group operation

The element is thus the neutral element of the group.

Examples

All trivial groups are isomorphic to each other. Examples of trivial groups are:

  • The cyclic group of degree
  • The alternating group of degree
  • The symmetric group of a singleton

Properties

  • Since the group operation is commutative, the trivial group is an abelian group.
  • The only subgroup of the trivial group is the trivial group itself
  • The trivial group is generated by the empty set. Here the empty product yields after the usual convention, the neutral element.
  • Each group contains the trivial group and itself ( trivial ) as a normal subgroup. The trivial group is therefore usually not considered as a simple group.
  • In the category of groups Grp the trivial group acts as a null object.
Trivial ring Wolfram Research
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