Prove that there is no simple group of order n, where 201 ≤n ≤ 235 and n is not prime.
Prove that there is no simple group of order n, where 201 ≤
n ≤ 235 and n is not prime.
We have to prove that there is no simple group of order , where and is not prime.
First of all we will prove that a group of order , where is prime and is not a simple group.
If then order is and it is a Abelian group.
If then is the only divisor of that is equal to .
Hence, by Sylow test for non simplicity, the group is no simple.
Therefore, we only need to study group of order .
Now,
Hence, by Sylow test for non simplicity, we see that a group of order is not simple.
Similarly, a group of order is not simple.
Now, we will discuss only groups of order .
Let be a group of order
Hence,
Using odd test, we can conclude that a group of order cannot be a simple group.
Therefore, there is no simple group of order .
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