Trivial group

From Maths
Jump to: navigation, search
For other uses of trivial see the page trivial

Definition

Let [ilmath]G:\eq\{e\} [/ilmath], the set containing one object, which we shall call [ilmath]e[/ilmath], and consider the binary operation given by the function: [ilmath]*:G\times G\rightarrow G[/ilmath] given by [ilmath]*:(e,e)\mapsto e[/ilmath], then we claim:

  • [ilmath](\{e\},*)[/ilmath] is a group

This is the trivial group, any group isomorphic to the trivial group is also said to be trivial.

We use [ilmath]e[/ilmath] for the object as it is the identity element of the group


Claims:

  1. This is indeed a group
  2. This is an Abelian group (the operation is commutative)
  3. [ilmath]e[/ilmath] is the identity element of the group.

Notations

  • When dealing with Abelian groups we may write the trivial group as [ilmath]0[/ilmath], as [ilmath]0[/ilmath] the common way to write the identity of any Abelian group
  • When dealing with groups in general (that are not or need not be commutative) we use [ilmath]1[/ilmath] for the trivial group, as the identity - in multiplicative notation - is often written [ilmath]1[/ilmath]
  • Sometimes we will write [ilmath]e[/ilmath] if it would be ambiguous to use [ilmath]0[/ilmath] or {{M|1]}.

It is a slight abuse of notation to identify the group with its only element, but this is in line with other uses, for example [ilmath]0[/ilmath] is commonly used for the trivial group homomorphism that sends everything to the identity element of the co-domain group.

Proof of claims

Grade: D
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:
Important but really easy

This proof has been marked as an page requiring an easy proof

References

Grade: D
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
References not needed