Tour:Trivial group

This article adapts material from the main article: trivial group

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WHAT YOU NEED TO DO:

• Understand the definition of trivial group given below
• Convince yourself that this is indeed a group

Definition

Verbal definition

The trivial group is the group with only one element, which is its identity element. The trivial group is usually denoted as $1$, $\{ 1 \}$, or $\{ e \}$.

PONDER (WILL BE EXPLORED LATER IN THE TOUR):

• Over why the trivial group occurs as a subgroup in any group
• Over why the trivial group is Abelian
  

WHAT'S MORE: Some alternative descriptions, and important facts, about the trivial group. Ignore the parts that use terminology you haven't encountered so far.

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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

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.

This page is part of the Groupprops Guided tour for beginners (Jump to beginning of tour). If you found anything difficult or unclear, make a note of it; it is likely to be resolved by the end of the tour.
PREVIOUS: Subgroup | UP: Introduction one (beginners) | NEXT: Verifying the group axioms