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
 The cyclic group on one element
 The symmetric group on one element
 The projective general linear group of order 1 over any field
 The special linear group of order 1 over any field
 The general linear group
 The orthogonal group of order 1 over a field of characteristic two
Multiplication table
Element  (identity element) 

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 oneelement 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.