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September 17th, 2009, 09:05 PM  #1 
Member Joined: Aug 2008 Posts: 84 Thanks: 0  Normal sylow subgroup of a group of order p^aq^b
So this is a curious little question, where p and q are of course primes. Not quite sure how to go about it. Clearly the case p = q is of no interest. I think my current repertoire of techniques in group theory isn't enough to handle this problem. Perhaps sections one and two of dummit and foote chapter 6 might shed some light. Appreciate any helpful hints, this problem has been annoying me.

September 19th, 2009, 07:02 PM  #2 
Senior Member Joined: Dec 2008 Posts: 160 Thanks: 0  Re: Normal sylow subgroup of a group of order p^aq^b
It can be shown that Sylow psubgroup is normal if and only if it is the only Sylow psubgroup. It requires that is not multiple of and is not multiple of In this case our group is a direct product of two Sylow groups. 
September 20th, 2009, 12:03 PM  #3 
Member Joined: Aug 2008 Posts: 84 Thanks: 0  Re: Normal sylow subgroup of a group of order p^aq^b
So you're saying that we must impose the conditions that b is not a multiple of p1 and a is not a multiple of q1 for this to be true? The original question is verbatim from a qualifying exam.

September 21st, 2009, 10:53 AM  #4 
Senior Member Joined: Dec 2008 Posts: 160 Thanks: 0  Re: Normal sylow subgroup of a group of order p^aq^b
Conditions are coming from the fact that it has to be only one pSylow subgroup and only one qSylow subgroup. You can check Sylow theorem II, first and second statements of it. P.S. I failed number qualifying exams in the past. 
September 21st, 2009, 01:53 PM  #5 
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Normal sylow subgroup of a group of order p^aq^b
How does (q1)a relate to the uniqueness of a Sylowp subgroup? There's probably some blindingly obvious number theory I'm missing, but I don't see the relation between the 2 ideas. 
September 21st, 2009, 05:06 PM  #6 
Senior Member Joined: Dec 2008 Posts: 160 Thanks: 0  Re: Normal sylow subgroup of a group of order p^aq^b
Here are full path: Sylow theorem states that if G is finite group and with m not divisible by , then: (a) Any two Sylow psubgroups are conjugate. (b) Let be the number of Sylow psubgroups in G; then and divides (c) Every psubgroup of G is contained in a Sylow psubgroup To be the only subgroup, must be , in our case , so . Order of in is , so (here is my correction) if , then exists c so that may not be 1. Similar thoughts about q. Overall Now If N is the only subgroup, it is clearly normal (actually, it is also a characteristic subgroup). Otherwise, if N is normal (a) of Sylow theorem states that is the only one. 
September 21st, 2009, 09:52 PM  #7  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Normal sylow subgroup of a group of order p^aq^b Quote:
(Sorry for hijacking this thread)  

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group, normal, order, paqb, subgroup, sylow 
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