Answered: Prove that there is no simple group of…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 35E: Exercises 35. Prove that any two groups of order are isomorphic.

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

Expert Solution

Step 1

We have to prove that there is no simple group of order $n$ , where $201 \leq n \leq 235$ and $n$ is not prime.

First of all we will prove that a group of order $m q$ , where $q$ is prime and $m \leq q$ is not a simple group.

If $m = q$ then order is $q^{2}$ and it is a Abelian group.

If $m < q$ then $1$ is the only divisor of $p q$ that is equal to $1 (m o d q)$ .

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,

$\begin{matrix} 208 = 2^{4} \times 13 \\ 210 = 2 \times 3 \times 5 \times 7 \\ 216 = 2^{3} \times 3^{3} \\ 220 = 2^{2} \times 5 \times 11 \\ 224 = 2^{5} \times 7 \\ 231 = 3 \times 7 \times 11 \end{matrix}$

Hence, by Sylow test for non simplicity, we see that a group of order $2^{4} \times 13$ is not simple.

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

Now, we will discuss only groups of order $210 o r 216$ .

Let $G$ be a group of order $210$

Hence, $210 = 2 \times (3 \times 5 \times 7)$

Using $2 \times$ 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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Prove that there is no simple group of order n, where 201 ≤n ≤ 235 and n is not prime.