Trivial Group is Cyclic Group

Theorem
The trivial group is a cyclic group and therefore abelian.

Proof
In Trivial Group is Group it is shown that the algebraic structure $\struct {\set e, \circ}$ such that $e \circ e = e$ is in fact a group.

It remains to be shown that it is cyclic.

In order for $G$ to be a cyclic group, every element $x$ of $G$ has to be expressible in the form $x = g^n$ for some $g \in G$ and some $n \in \Z$.

In this case, for every integer $n$, every element of $G$ can be expressed in the form $e^n$.

Thus $G$ is trivially a cyclic group.

Thus $G$ is abelian from Cyclic Group is Abelian.