Trivial Quotient Group is Quotient Group

Theorem
Let $G$ be a group.

Then:
 * $G / \left\{{e_G}\right\} \cong G$

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

The quotient group $G / \left\{{e_G}\right\}$ is known as the trivial quotient group.

Proof

 * $\left\{{e_G}\right\} \triangleleft G$ from Trivial Subgroup 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 a group isomorphism and therefore:
 * $G / \left\{{e_G}\right\} \cong G$