Homomorphisms, Kernels & Quotient Groups
Homomorphisms, Kernels & Quotient Groups
A group homomorphism $\phi:G\to H$ preserves the group operation: $\phi(ab)=\phi(a)\phi(b)$. The kernel $\ker\phi=\{g:\phi(g)=e_H\}$ is always a normal subgroup; the image $\text{Im}\phi$ is a subgroup of $H$. The First Isomorphism Theorem — $G/\ker\phi\cong\text{Im}\phi$ — is the central structure theorem, connecting quotient groups to images of homomorphisms.
Normal Subgroups & Quotient Groups
$N\trianglelefteq G$ (normal subgroup) if $gNg^{-1}=N$ for all $g\in G$. Equivalently: left and right cosets coincide. The quotient group $G/N=\{gN:g\in G\}$ with operation $(aN)(bN)=(ab)N$ is well-defined iff $N$ is normal. Examples: $n\mathbb{Z}\trianglelefteq\mathbb{Z}$; $\mathbb{Z}/n\mathbb{Z}=\mathbb{Z}_n$. Every subgroup of an abelian group is normal. The center $Z(G)=\{g:\forall h,\,gh=hg\}$ is always normal.
First Isomorphism Theorem
If $\phi:G\to H$ is a group homomorphism, then $\ker\phi\trianglelefteq G$ and $$G/\ker\phi\cong\text{Im}(\phi).$$ The isomorphism is $\bar\phi:G/\ker\phi\to\text{Im}\phi$, $g\ker\phi\mapsto\phi(g)$. Second: if $H\leq G$ and $N\trianglelefteq G$, then $HN/N\cong H/(H\cap N)$. Third: if $N\trianglelefteq M\trianglelefteq G$ (with $N\trianglelefteq G$), then $(G/N)/(M/N)\cong G/M$.
Example 1
Use the First Isomorphism Theorem to show $\mathbb{Z}/n\mathbb{Z}\cong\mathbb{Z}_n$.
Solution: Define $\phi:\mathbb{Z}\to\mathbb{Z}_n$ by $\phi(k)=k\bmod n$. $\phi$ is a surjective homomorphism (it preserves addition). $\ker\phi=\{k:k\equiv 0\pmod n\}=n\mathbb{Z}$. By First Isomorphism Theorem: $\mathbb{Z}/n\mathbb{Z}\cong\text{Im}\phi=\mathbb{Z}_n$. ✓
Example 2
Show $\mathbb{R}^*/\{\pm 1\}\cong\mathbb{R}^+$ (positive reals under multiplication).
Solution: $\phi:\mathbb{R}^*\to\mathbb{R}^+$, $\phi(x)=|x|$. Surjective homomorphism ($\phi(xy)=|xy|=|x||y|$). $\ker\phi=\{x:|x|=1\}=\{1,-1\}$. By FIT: $\mathbb{R}^*/\{\pm 1\}\cong\mathbb{R}^+$.
Practice
- Prove $\ker\phi$ is a normal subgroup for any homomorphism $\phi$.
- Show $D_4/Z(D_4)\cong\mathbb{Z}_2\times\mathbb{Z}_2$ (dihedral group of square modulo center).
- Prove the correspondence theorem: subgroups of $G/N$ correspond to subgroups of $G$ containing $N$.
- Show that if $[G:H]=2$ then $H\trianglelefteq G$.
Show Answer Key
1. Let $k\in\ker\phi$ and $g\in G$. Then $\phi(gkg^{-1})=\phi(g)\phi(k)\phi(g^{-1})=\phi(g)e'\phi(g)^{-1}=e'$. So $gkg^{-1}\in\ker\phi$, hence $\ker\phi\trianglelefteq G$.
2. $D_4=\{e,r,r^2,r^3,s,sr,sr^2,sr^3\}$, $Z(D_4)=\{e,r^2\}$. $D_4/Z(D_4)$ has order 4. Every element has order 2 in the quotient (since $(rZ)^2=r^2Z=Z$), so $D_4/Z(D_4)\cong\mathbb{Z}_2\times\mathbb{Z}_2$.
3. There is a bijection between subgroups of $G/N$ and subgroups of $G$ containing $N$, given by $H/N\leftrightarrow H$ (for $N\le H\le G$). This preserves normality and indices.
4. If $[G:H]=2$, then $G/H=\{H,gH\}$ for any $g\notin H$. For any $g\in G$: if $g\in H$, $gH=Hg=H$. If $g\notin H$, $gH=G\setminus H=Hg$. So left and right cosets coincide, meaning $H\trianglelefteq G$.