Second Isomorphism Theorem/Groups

Theorem

Let $G$ be a group, and let:

• $H$ be a subgroup of $G$.
• $N$ be a normal subgroup of $G$.

Then:

$\displaystyle \frac H {H \cap N} \cong \frac {H N} N$

where $\cong$ denotes group isomorphism.

This result is also referred to by some sources as the first isomorphism theorem.

Proof

• From Quotient Group, for $G / H$ to be defined, it is necessary for $H \triangleleft G$.

The fact that Intersection with Normal Subgroup is Normal gives us that $N \cap H \triangleleft H$.

Also, $N \triangleleft N H = \left \langle {H, N} \right \rangle$ follows from Subgroup Product with Normal Subgroup as Generator.

• Now we define a mapping $\phi: H \to H N / N$ by the rule $\phi \left({h}\right) = h N$.

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 $\phi \left({x y}\right) = x y N = \left({x N}\right) \left({y N}\right) = \phi \left({x}\right) \phi \left({y}\right)$.

Then:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \ker \left({\phi}\right)\)	\(=\)	\(\displaystyle \left\{ {h \in H: \phi \left({h}\right) = e_{H N / N} }\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {h \in H: h N = N}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {h \in H: h \in N}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle H \cap N\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Then we see that $\phi$ is a surjection because $h n N = h N \in H N / N$ is $\phi \left({h}\right)$.

The result follows from the First Isomorphism Theorem.

$\blacksquare$

Sources

• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$: Problem $\text{HH}$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 69$
• John F. Humphreys: A Course in Group Theory (1996): $\S 8$: Theorem $8.15$