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 Post subject: normal p-subgroups of a finite groupPosted: Fri, 17 Aug 2012 19:44:48 UTC
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Let be a finite group. Let and be normal -subgroups of for some prime with being a proper subgroup of and is elementary abelian p-group. Let be a normal subgroup of with . If contain every element of order of and, then contain every element of order of .

I need to prove the above statement. Thanks in advance.

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 Post subject: Re: normal p-subgroups of a finite groupPosted: Fri, 17 Aug 2012 20:34:14 UTC
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I'm a bit confused. Index is multiplicative, but if H has index p in K how can H < S and not H=S? Or perhaps you mean something besides the cyclic group of order p for "elementary abelian p-group". If so, please clarify.

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 Post subject: Re: normal p-subgroups of a finite groupPosted: Fri, 17 Aug 2012 20:42:02 UTC
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You are right and to correct that I am adding that is not of prime order.

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 Post subject: Re: normal p-subgroups of a finite groupPosted: Fri, 17 Aug 2012 20:46:00 UTC
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moont14263 wrote:
You are right and to correct that I am adding that is not of prime order.

Ah, never mind, I had forgotten my terminology, elementary just means "cyclic", not necessarily "of prime order". I think I understand what you're asking now.

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 Post subject: Re: normal p-subgroups of a finite groupPosted: Fri, 17 Aug 2012 21:00:02 UTC
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No elementary does not mean cyclic.
N is elementary abelian p-group if N is abelian and x^{p}=1 for all x in N.

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 Post subject: Re: normal p-subgroups of a finite groupPosted: Fri, 17 Aug 2012 21:03:27 UTC
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moont14263 wrote:
No elementary does not mean cyclic.
N is elementary abelian p-group if N is abelian and x^{p}=1 for all x in N.

Oh I see. So "elementary abelian p-group" means a finite abelian group of exponent p.

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 Post subject: Re: normal p-subgroups of a finite groupPosted: Fri, 17 Aug 2012 21:12:10 UTC
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I want to add that G is solvable and K/H is a chief factor of G.

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 Post subject: Re: normal p-subgroups of a finite groupPosted: Sat, 18 Aug 2012 20:01:51 UTC
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I got my question from the shaded area in this image http://tinypic.com/view.php?pic=vpajic&s=6

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