Trivial group

For other uses of trivial see the page trivial

Definition

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

• $(\{e\},*)$ is a group

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

We use $e$ 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. $e$ is the identity element of the group.

Notations

• When dealing with Abelian groups we may write the trivial group as $0$, as $0$ 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 $1$ for the trivial group, as the identity - in multiplicative notation - is often written $1$
• Sometimes we will write $e$ if it would be ambiguous to use $0$ 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 $0$ is commonly used for the trivial group homomorphism that sends everything to the identity element of the co-domain group.

Proof of claims

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References

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