## 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.

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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. $|G/K|=p$.

Then $p=p[H:K] \implies [H:K]=1$, i.e. H=K. Thus, H is normal in G.

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### 3 Responses to Singapore Haze & Subgroup of Smallest Prime Index

1. Joseph Nebus says:

I’m shocked to hear how bad the haze has gotten.

Liked by 1 person

• Yeah, it is pretty bad, my lungs hurt after breathing in the air.

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