Trivial group

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.

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References

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