normal p-subgroups of a finite group

moont14263 · **Joined:** Thu, 28 Jun 2012 01:54:37 UTC **Posts:** 19

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 $K=<S^{g},g \in G>$ , then

contain every element of order

of

.

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

Shadow · **Posted:** Fri, 17 Aug 2012 20:34:14 UTC

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.

moont14263 · **Joined:** Thu, 28 Jun 2012 01:54:37 UTC **Posts:** 19

You are right and to correct that I am adding that K/H

is not of prime order.

Shadow · **Posted:** Fri, 17 Aug 2012 20:46:00 UTC

moont14263 wrote:

You are right and to correct that I am adding that K/H

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.

moont14263 · **Joined:** Thu, 28 Jun 2012 01:54:37 UTC **Posts:** 19

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.

Shadow · **Posted:** Fri, 17 Aug 2012 21:03:27 UTC

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.

moont14263 · **Joined:** Thu, 28 Jun 2012 01:54:37 UTC **Posts:** 19

I want to add that G is solvable and K/H is a chief factor of G.

moont14263 · **Joined:** Thu, 28 Jun 2012 01:54:37 UTC **Posts:** 19

I got my question from the shaded area in this image http://tinypic.com/view.php?pic=vpajic&s=6

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normal p-subgroups of a finite group

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