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
Section: Chapter Questions
Problem 1RQ
icon
Related questions
Question

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 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.

 

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Groups
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,