Group of order 56 is not simple + Affordable Air Purifier

Haze strikes Singapore again, this time it is quite serious as the PSI is often above 300. For those seeking an affordable air purifier, do consider this air purifier which is one of the few that costs below $100. It is definitely reusable next year as the haze problem is not going to be solved in the near future.

Let G be a group of order 56. Show that G is not simple.

Proof:

We will use Sylow’s Theorem to show that either the 2-Sylow subgroup or 7-Sylow subgroup is normal.

$latex |G|=2^3cdot 7$

By Sylow’s Theorem $latex n_2mid 7, n_2equiv 1pmod 2$. Thus $latex n_2=1,7$.

Also, $latex n_7mid 8, n_7equiv 1pmod 7$. Therefore $latex n_7=1, 8$.

If $latex n_2=1$ or $latex n_7=1$, we are done…