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:

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

$\Box$

G1: Associativity

$\circ$ is associative:

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

trivially.

$\Box$

G2: Identity Element

$e$ is the identity:

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

$\Box$

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:

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

$\blacksquare$