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March 12th, 2011, 08:08 AM  #1 
Newbie Joined: Mar 2011 Posts: 26 Thanks: 0  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 psubgroup for some prime number p. (without using the characteristic group, hint  you can look at a psylow 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. Thanks 
March 12th, 2011, 02:10 PM  #2 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  Re: Minimal Normal subgroup of a solvable group
Okay, here's a way to do it for #2: Assume that H is not a pgroup, and consider a kSylow subgroup K of H. Since all Sylow subgroups of H are conjugated and H is abelian, K is the unique kSylow subgroup of H, and K is different from H (since K is not a pgroup). 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 pSylow of H, and is therefore equal to K. 
March 12th, 2011, 04:30 PM  #3  
Newbie Joined: Mar 2011 Posts: 26 Thanks: 0  Re: Minimal Normal subgroup of a solvable group Quote:
Thanks.  
March 13th, 2011, 01:31 AM  #4 
Site Founder Joined: Nov 2006 From: France Posts: 824 Thanks: 7  Re: Minimal Normal subgroup of a solvable group
They are the same group, since H has a unique pSylow ...


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