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.

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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.

By Sylow’s Theorem . Thus .

Also, . Therefore .

If or , we are done, as one of the Sylow subgroups is normal.

Suppose to the contrary and .

Number of elements of order 7 = 8 x (7-1)=48

Remaining elements = 56-48=8. This is just enough for one 2-Sylow subgroup, thus . This is a contradiction.

Therefore, a group of order 56 is simple.