Trivial group

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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 $GL(1,2)$
• The orthogonal group of order 1 over a field of characteristic two

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.