Trivial Quotient Group is Quotient Group

Theorem
Let $$G$$ be a group.

Then $$G / \left\{{e_G}\right\} \cong G$$.

Proof

 * $$\left\{{e_G}\right\} \triangleleft G$$ from Trivial Group and Group Itself are Normal.

Let $$x \in G$$. Then $$x \left\{{e_G}\right\} = \left\{{x e_G}\right\} = \left\{{x}\right\}$$.

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


 * Now we set up the natural epimorphism $$\psi: G \to G / \left\{{e_G}\right\}$$:

$$\forall x \in G: \phi \left({x}\right) = x \left\{{e_G}\right\}$$

which is of course a surjection.

We now need to establish that it is an injection.

Let $$p, q \in G$$.

$$ $$ $$ $$

So $$\psi$$ is an isomorphism and therefore $$G / \left\{{e_G}\right\} \cong G$$.