Difference between revisions of "Trivial group"

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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Verbal definition

The trivial group is the group with only one element, which is its identity element.

Alternative definitions

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.