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.

[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


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


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

|G|=2^3\cdot 7

By Sylow’s Theorem n_2\mid 7, n_2\equiv 1\pmod 2. Thus n_2=1,7.

Also, n_7\mid 8, n_7\equiv 1\pmod 7. Therefore n_7=1, 8.

If n_2=1 or n_7=1, we are done, as one of the Sylow subgroups is normal.

Suppose to the contrary n_2=7 and n_7=8.

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 n_2=1. This is a contradiction.

Therefore, a group of order 56 is simple.

Author: mathtuition88

One thought on “Group of order 56 is not simple + Affordable Air Purifier”

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.