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September 17th, 2009, 09:05 PM   #1
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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.
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September 19th, 2009, 07:02 PM   #2
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Re: Normal sylow subgroup of a group of order p^aq^b

It can be shown that Sylow p-subgroup is normal if and only if it is the only Sylow p-subgroup.
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.
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September 20th, 2009, 12:03 PM   #3
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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 p-1 and a is not a multiple of q-1 for this to be true? The original question is verbatim from a qualifying exam.
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September 21st, 2009, 10:53 AM   #4
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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 p-Sylow subgroup and only one q-Sylow subgroup.
You can check Sylow theorem II, first and second statements of it.
P.S.
I failed number qualifying exams in the past.
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September 21st, 2009, 01:53 PM   #5
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Re: Normal sylow subgroup of a group of order p^aq^b

How does (q-1)|a relate to the uniqueness of a Sylow-p subgroup?
There's probably some blindingly obvious number theory I'm missing, but I don't see the relation between the 2 ideas.
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September 21st, 2009, 05:06 PM   #6
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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 p-subgroups are conjugate.
(b) Let be the number of Sylow p-subgroups in G; then and divides
(c) Every p-subgroup of G is contained in a Sylow p-subgroup

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.
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September 21st, 2009, 09:52 PM   #7
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Re: Normal sylow subgroup of a group of order p^aq^b

Quote:
Originally Posted by zolden
Order of in is , so (here is my correction) if , then exists c so that may not be 1.
Ah! That is what I was missing. I'm not completely sure I understand the last step, but I need to sit down and work it out-- I "believe" you, now, though.

(Sorry for hijacking this thread)
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