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March 19th, 2015, 04:48 AM  #1 
Newbie Joined: Mar 2015 From: Chennai Posts: 2 Thanks: 0  how to find group and subgroup
how to find simple group of order 168. what is the number of subgroups of G of order 7? Thanks in advance 
March 19th, 2015, 04:51 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
I am puzzled by this. If, as you appear to be saying, you know nothing about "simple groups", where did you get this problem? I, frankly, don't know how to prove it, but Wikipedia says "The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and it is possible to prove that every simple group of order 168 is isomorphic to PSL" It is easy to see that the prime factorization of 168 is $\displaystyle 2^2(3)(7)$. Notice that 7 is a prime factor of 168. 
March 19th, 2015, 07:11 PM  #3 
Newbie Joined: Mar 2015 From: Chennai Posts: 2 Thanks: 0 
Thanks for your response. I am preparing for national lectureship exam; this was previously asked question from abstract algebra. From your solution, I noticed that prime factor should not be considered (168 = 2 cube (3) (7) ). In that case the answer would be 8 .

May 10th, 2015, 03:09 AM  #4  
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra  Quote:
Itâ€™s $2^3(3)(7)$.  

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