Second Isomorphism Theorem/Groups
Theorem
Let $G$ be a group, and let:
 $(1): \quad H$ be a subgroup of $G$
 $(2): \quad N$ be a normal subgroup of $G$.
Then:
 $\dfrac H {H \cap N} \cong \dfrac {H N} N$
where $\cong$ denotes group isomorphism.
Proof
The fact that $N$ is normal, together with Intersection with Normal Subgroup is Normal, gives us that $N \cap H \lhd H$.
Also, $N \lhd N H = \gen {H, N}$ follows from Subset Product with Normal Subgroup as Generator.
Now we define a mapping $\phi: H \to H N / N$ by the rule:
 $\map \phi h = h N$
Note that $N$ need not be a subset of $H$.
Therefore, the coset $h N$ is an element of $H N / N$ rather than of $H / N$.
Then $\phi$ is a homomorphism, as:
 $\map \phi {x y} = x y N = \paren {x N} \paren {y N} = \map \phi x \map \phi y$
Then:
\(\ds \map \ker \phi\)  \(=\)  \(\ds \set {h \in H: \map \phi h = e_{H N / N} }\)  
\(\ds \)  \(=\)  \(\ds \set {h \in H: h N = N}\) 


\(\ds \)  \(=\)  \(\ds \set {h \in H: h \in N}\)  
\(\ds \)  \(=\)  \(\ds H \cap N\) 
Then we see that $\phi$ is a surjection because $h n N = h N \in H N / N$ is $\map \phi h$.
The result follows from the First Isomorphism Theorem.
$\blacksquare$
Also known as
This result is also referred to by some sources as the first isomorphism theorem.
Also see
Sources
 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.16$
 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): $1$: Subgroups: $1.\text{T}.5$
 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Problem $\text{HH}$
 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 69$. The Second Isomorphism Theorem
 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Theorem $8.15$