Isomorphism theorems

Three foundational results relating quotients, Homomorphism, and subgroup structure.

First Isomorphism Theorem

First Isomorphism Theorem

Let $\theta :G\to H$ be a Homomorphism. Then

$$\boxed{G/\ker (\theta )\cong \mathrm{Im}(\theta ).}$$

Mnemonic. “Quotient by the kernel = image.”

Why it matters. Every surjective homomorphism $G\to H$ is essentially a Quotient group map: $H\cong G/\ker (\theta )$. Quotients and images are two views of the same thing.

Second Isomorphism Theorem

Second Isomorphism Theorem

Let $G$ be a group, $H\le G$, $N⊴G$. Then

$$\boxed{H/(H\cap N)\cong (HN)/N.}$$

Picture. Mod out $H$ by the part of $H$ lying in $N$, and you get the same as taking the larger group $HN$ and modding by $N$.

Third Isomorphism Theorem

Third Isomorphism Theorem

Let $G$ be a group with $N⊴G$ and $H⊴G$ with $N\le H$. Then $H/N⊴G/N$, and

$$\boxed{(G/N)/(H/N)\cong G/H.}$$

Mnemonic. “Quotients telescope” - taking a quotient by $N$ then by $H/N$ is the same as taking a quotient by $H$ directly.

Worked Application

For $G={\mathrm{GL}}_2(\mathbb{R})$, $K={\mathrm{SL}}_2(\mathbb{R})$, $P=\{M:\det (M)>0\}$:

• $G/K\cong {\mathbb{R}}^{\ast }$ (First Iso, applied to $\det$).
• $P/K\cong {\mathbb{R}}_{>0}$.
• $G/P\cong C_2$.
• Third Iso: $(G/K)/(P/K)\cong G/P$, i.e. ${\mathbb{R}}^{\ast }/{\mathbb{R}}_{>0}\cong C_2$. ✓

Significance

Together, the three theorems describe how subgroup lattices and quotient lattices interact. They reduce many group-theoretic questions to a sequence of “factor and identify” steps.