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March 12th, 2011, 07:08 AM   #1
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Minimal Normal subgroup of a solvable group

Let be a finite solvable group and a minimal normal subgroup of .

1. Show is abelian
2. Show is a p-subgroup for some prime number p. (without using the characteristic group, hint - you can look at a p-sylow group of ).
3. Let show that .
4. Show that is a direct sum of cyclic grouops of order p.

I was able to show 1. Currently stuck on #2, need help there.

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March 12th, 2011, 01:10 PM   #2
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Re: Minimal Normal subgroup of a solvable group

Okay, here's a way to do it for #2:
Assume that H is not a p-group, and consider a k-Sylow subgroup K of H. Since all Sylow subgroups of H are conjugated and H is abelian, K is the unique k-Sylow subgroup of H, and K is different from H (since K is not a p-group). If we manage to prove that K is normal in G then we will be done since then H will not be a minimal normal subgroup of G (absurd).
Proof: considering an element a in G,

is a p-Sylow of H, and is therefore equal to K.
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March 12th, 2011, 03:30 PM   #3
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Re: Minimal Normal subgroup of a solvable group

Quote:
Originally Posted by julien
Proof: considering an element a in G,

is a p-Sylow of H, and is therefore equal to K.
did you mean is a k-sylow subgroup of H? I think I understand everything else

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March 13th, 2011, 12:31 AM   #4
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Re: Minimal Normal subgroup of a solvable group

They are the same group, since H has a unique p-Sylow ...
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