Normal subgroup

A Subgroup $H\le G$ is normal in $G$, written $H⊴G$, if any of these equivalent conditions hold:

1. $g^{-1}Hg=H$ for all $g\in G$ (conjugation-invariance).
2. $g^{-1}hg\in H$ for all $g\in G$ and $h\in H$.
3. $gH=Hg$ for all $g\in G$ (left and right Coset agree).
4. $H$ is the Kernel of some homomorphism $\theta :G\to (\text{some group})$.

Why It Matters

Only normal subgroups produce well-defined quotient groups $G/H$ (see Quotient group). Without normality, the natural multiplication $(g_{1}H)(g_{2}H)=(g_1{g}_2)H$ depends on the representatives chosen.

Detection

• Index 2: any subgroup of index 2 is automatically normal. (Two cosets, $H$ and the rest, must agree on left and right.)
• Unique Sylow’s theorems: if $a(p)=1$, the unique Sylow $p$-subgroup is normal (conjugation must send it to itself).
• Centre: $Z(G)=\{z\in G:gz=zg\forall g\}$ is always normal.
• Kernels: every kernel of a homomorphism is normal - and conversely, every normal subgroup is a kernel.

Examples

• $A_n⊴S_n$ - kernel of the Signature.
• ${\mathrm{SL}}_n(\mathbb{R})⊴{\mathrm{GL}}_n(\mathbb{R})$ - kernel of $\det$.
• In an abelian group, every subgroup is normal.
• $V_4=\{e,(12)(34),(13)(24),(14)(23)\}⊴A_4$ - important for $A_4$‘s structure.

Non-Examples

• $⟨(12)⟩/⊴S_3$: $(13)(12)(13{)}^{-1}=(23)\notin ⟨(12)⟩$.

Constructing Quotient Groups

If $H⊴G$, the Coset form the Quotient group $G/H$ under $(g_{1}H)(g_{2}H)=(g_1{g}_2)H$. The natural map $G\to G/H$, $g\mapsto gH$, is a surjective homomorphism with kernel $H$.