Quotient of Group by Itself

Theorem
Let $G$ be a group.

Let $G / G$ be the quotient group of $G$ by itself.

Then:
 * $G / G \cong \set e$

That is, the quotient of a group by itself is isomorphic to the trivial group.

Proof
Let the homomorphism $\phi: G \to \set e$ be defined as:


 * $\forall g \in G: \map \phi g = e$

Then:
 * $\map \ker \phi = G$

and:
 * $\Img \phi = \set e$

By the First Isomorphism Theorem:
 * $G / \map \ker \phi \cong \Img \phi$

Hence the result:
 * $G / G \cong \set e$