Abstract Algebra Abstract Algebra Math Forum

 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 p-group and H
 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 itself--this is not hard to see.
December 12th, 2008, 12:50 AM   #4
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 Originally Posted by Spartan Math (3) is true in general (for any group). H always normalizes itself--this is not hard to see.
im new in this area so i do not understand easily.. may u pls explain detailly.

December 12th, 2008, 12:51 AM   #5
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 Originally Posted by pseudonym 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.
thank you for ur attention... it s really help me to understand how ? can solve the quest.

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