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December 10th, 2008, 07:57 AM  #1 
Newbie Joined: Dec 2008 Posts: 9 Thanks: 0  pls quickly answer ma quest about sylow thm 1. Let p be prime, and G be a finite group. If every element of G has order a power of p, then G = p^n for some n?0. (Hint: Use Cauchy’s theorem.) 2. Tell as much as possible about the subgroups of a group of order 30 and of a group of order 40. 3. Let G be a finite pgroup and H<G. Prove that H<N G(H) . please help me quickly... ? dont understand these ques.s 
December 10th, 2008, 10:04 AM  #2 
Senior Member Joined: Nov 2008 Posts: 199 Thanks: 0  Re: pls quickly answer ma quest about sylow thm
for 1 suppose the order (size) of G is not a power of p. Then there must exist a prime q dividing the order of G such that q is not equal to p (by the fundamental theorem of arithmetic). By cauchy's theorem this means G has an element with order q. This is a contradiction as we are told that every element of G has order a power of p. As q is a prime distinct from p this cannot happen.

December 10th, 2008, 02:58 PM  #3 
Member Joined: Aug 2008 Posts: 84 Thanks: 0  Re: pls quickly answer ma quest about sylow thm
(3) is true in general (for any group). H always normalizes itselfthis is not hard to see.

December 12th, 2008, 12:50 AM  #4  
Newbie Joined: Dec 2008 Posts: 9 Thanks: 0  Re: pls quickly answer ma quest about sylow thm Quote:
 
December 12th, 2008, 12:51 AM  #5  
Newbie Joined: Dec 2008 Posts: 9 Thanks: 0  Re: pls quickly answer ma quest about sylow thm Quote:
 

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