Trivial Quotient Group is Quotient Group

Theorem
Let $G$ be a group.

Then the trivial quotient group:
 * $G / \set {e_G} \cong G$

where:
 * $\cong$ denotes group isomorphism
 * $e_G$ denotes the identity element of $G$

is a quotient group.

Proof
From Trivial Subgroup is Normal:
 * $\set {e_G} \lhd G$

Let $x \in G$.

Then:
 * $x \set {e_G} = \set {x e_G} = \set x$

So each (left) coset of $G$ modulo $\set {e_G}$ has one element.

Now we set up the quotient epimorphism $\psi: G \to G / \set {e_G}$:


 * $\forall x \in G: \map \phi x = x \set {e_G}$

which is of course a surjection.

We now need to establish that it is an injection.

Let $p, q \in G$.

So $\psi$ is a group isomorphism and therefore:
 * $G / \set {e_G} \cong G$