Trivial Group is Cyclic Group

Theorem

The trivial group is a cyclic group.

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.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.4$. Cyclic groups: Example $100$
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$