# 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

$\set {e_G} \lhd G$

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 $p, q \in G$.

 $\displaystyle \map \phi p$ $=$ $\displaystyle \map \phi q$ $\displaystyle \leadsto \ \$ $\displaystyle p \set {e_G}$ $=$ $\displaystyle q \set {e_G}$ Definition of $\phi$ $\displaystyle \leadsto \ \$ $\displaystyle \set p$ $=$ $\displaystyle \set q$ from above $\displaystyle \leadsto \ \$ $\displaystyle p$ $=$ $\displaystyle q$ Definition of Set Equality

So $\psi$ is a group isomorphism and therefore:

$G / \set {e_G} \cong G$

$\blacksquare$