Trivial Group is Group

Theorem
The trivial group is a group.

Proof
Let $G = \struct {\set e, \circ}$ be an algebraic structure.

G0: Closure
For $G$ to be a group, it must be closed.

So it must be the case that:
 * $\forall e \in G: e \circ e = e$

G1: Associativity
$\circ$ is associative:


 * $e \circ \paren {e \circ e} = e = \paren {e \circ e} \circ e$

trivially.

G2: Identity Element
$e$ is the identity:


 * $\forall e \in G: e \circ e = e$

G3: Inverse Elements
Every element of $G$ (all one of them) has an inverse:

This follows from the fact that the Identity is Self-Inverse, and the only element of $G$ is indeed the identity:


 * $e \circ e = e \implies e^{-1} = e$