Trivial Subgroup is Subgroup

Theorem

Let $\struct {G, \circ}$ be a group whose identity is $e$.

Then the trivial subgroup $\struct {\set e, \circ}$ is indeed a subgroup of $\struct {G, \circ}$.

Proof

Using the One-Step Subgroup Test:

$(1): \quad e \in \set e \leadsto \set e \ne \O$
$(2): \quad e \in \set e \leadsto e \circ e^{-1} = e \in \set e$

$\blacksquare$