Trivial Group is Group

Theorem

The trivial group is a group.

Proof

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

Group Axiom $\text G 0$: Closure

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

So it must be the case that:

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

$\Box$

Group Axiom $\text G 1$: Associativity

$\circ$ is associative:

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

trivially.

$\Box$

Group Axiom $\text G 2$: Existence of Identity Element

$e$ is the identity:

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

$\Box$

Group Axiom $\text G 3$: Existence of Inverse Element

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:

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

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.5$. Examples of groups: Example $78$
• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.3$: Example $10$
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 26$