Prove that there is no simple group of order n, where 201 ≤n ≤ 235 and n is not prime.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Prove that there is no simple group of order n, where 201 ≤
n ≤ 235 and n is not prime.

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Step 1

We have to prove that there is no simple group of order n, where 201n235 and n is not prime.

First of all we will prove that a group of order mq, where q is prime and mq is not a simple group.

If m=q then order is q2 and it is a Abelian group.

If m<q then 1 is the only divisor of pq that is equal to 1mod q.

Hence, by Sylow test for non simplicity, the group is no simple.

Therefore, we only need to study group of order 208,210,216,220,224,231.

Now,

208=24×13210=2×3×5×7216=23×33220=22×5×11224=25×7231=3×7×11

Hence, by Sylow test for non simplicity, we see that a group of order 24×13 is not simple.

Similarly, a group of order 220, 224, 231 is not simple.

Now, we will discuss only groups of order 210 or 216.

Let G be a group of order 210

Hence, 210=2×3×5×7

Using 2× odd test, we can conclude that a group of order 210 cannot be a simple group.

Therefore, there is no simple group of order 210.

 

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