Singapore Haze & Subgroup of Smallest Prime Index

Recently, the Singapore Haze is getting quite bad, crossing the 200 PSI Mark on several occasions. Do consider purchasing a Air Purifier, or some N95 Masks, as the haze problem is probably staying for at least a month. Personally, I use Nasal Irrigation (Neilmed Sinus Rinse), which has tremendously helped my nose during this haze period. It can help clear out dust and mucus trapped in the nose.

[S$89.90][Clean Air]2015 Air Purifier Singapore brand and 1 year warranty with HEPA Activated Carbon UV-C germicidal killer lamp silent operation and high efficiency etc

WWW.QOO10.SG

[S$19.90]Haze Prevention~ Nasal Rinse™- Flush out mucous~germs~bacteria~and dirt internally. Clear blocked nose and very soothing.

WWW.QOO10.SG

Previously, we proved that any subgroup of index 2 is normal. It turns out that there is a generalisation of this theorem. Let $p$ be the smallest prime divisor of a group $G$ . Then, any subgroup $H\leq G$ of index $p$ is normal in $G$ .

Proof: Let $H$ be a subgroup of $G$ of index $p$ . Let $G$ act on the left cosets of $H$ by left multiplication: $\forall x\in G$ , $x\cdot gH=xgH$ .

This group action induces a group homomorphism $\phi:G\to S_p$ .

Let $K=\ker \phi$ . If $x\in K$ , then $xgH=gH$ for all $g\in G$ . In particular when g=1, xH=H, i.e. $x\in H$ .

Thus $K\subseteq H$ . In particular, $K\trianglelefteq H$ , since $\ker\phi$ is a normal subgroup of $G$ .

We have $G/K\cong \phi(G)\leq S_p$ . Thus $|G/K|\mid p!$ .

Also note that $|G|=|G/K||K|$ . Note that $|G/K|\neq 1$ since $|G/K|=[G:H][H:K]=p[H:K]\geq p$ .

Let $q$ be a prime divisor of $|G/K|$ . Then $q\leq p$ since $|G/K|\mid p!$ . Also, $q\mid |G|$ . Since $p$ is the smallest prime divisor of $|G|$ , $p\leq q$ . Therefore, $p=q$ , i.e. $|G/K|=p$ .