# Quotient of Group by Itself

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## 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:

$G / G \cong \set e$

$\blacksquare$