Quotient Epimorphism is Epimorphism/Group

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Let $G$ be a group.

Let $N$ be a normal subgroup of $G$.

Let $G / N$ be the quotient group of $G$ by $N$.

Let $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$:

$\forall x \in G: \map {q_N} x = x N$

Then $q_N$ is a group epimorphism whose kernel is $N$.


The proof follows from Quotient Mapping on Structure is Canonical Epimorphism.

When $N \lhd G$, we have:

\(\displaystyle \forall x, y \in G: \ \ \) \(\displaystyle \map {q_N} {x y}\) \(=\) \(\displaystyle x y N\) Definition of Quotient Group Epimorphism
\(\displaystyle \) \(=\) \(\displaystyle \paren {x N} \paren {y N}\) Definition of Quotient Group
\(\displaystyle \) \(=\) \(\displaystyle \map {q_N} x \map {q_N} y\) Definition of Quotient Group Epimorphism

Therefore $q$ is a homomorphism.

We have that:

$\forall x \in G: x N \in G / N = \map {q_N} x$

so $q$ is surjective.

Therefore $q$ is an epimorphism.

Let $x \in G$.

\(\displaystyle x\) \(\in\) \(\displaystyle \map \ker {q_N}\)
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle \map {q_N} x\) \(=\) \(\displaystyle e_{G/N}\) Definition of Kernel of Group Homomorphism
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x N\) \(=\) \(\displaystyle e_{G/N} N\) Definition of Quotient Group Epimorphism
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x N\) \(=\) \(\displaystyle N\) Coset by Identityā€ˇ
\(\displaystyle \leadstoandfrom \ \ \) \(\displaystyle x\) \(\in\) \(\displaystyle N\) Left Coset Equals Subgroup iff Element in Subgroup

thus proving that $\map \ker {q_N} = N$ from definition of subset.



In Kernel is Normal Subgroup of Domain it was shown that the kernel of a group homomorphism is a normal subgroup of its domain.

In this result it has been shown that every normal subgroup is a kernel of at least one group homomorphism of the group of which it is the subgroup.

We see that when a subgroup is normal, its cosets make a group using the group operation defined as in this result.

However, it is not possible to make the left or right cosets of a non-normal subgroup into a group using the same sort of group operation.

Otherwise there would be a group homomorphism with that subgroup as the kernel, and we have seen that this can not be done unless the subgroup is normal.