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May 27th, 2014, 12:31 AM  #1 
Newbie Joined: May 2014 From: Montreal Posts: 1 Thanks: 0  Abstract Algebra Sylow Subgroup
I have a question about abstract algebra, so if someone could help me answering this question please ... Suppose P,P' are 3Sylow subgroup, and let Q be their intersection and N the normalizer of Q. Problem: Explain why is the order of N divisible by 9 ? Thanks for your help. Regards, Last edited by skipjack; June 4th, 2014 at 08:00 AM. 
June 3rd, 2014, 03:46 PM  #2 
Senior Member Joined: Apr 2014 From: Greater London, England, UK Posts: 320 Thanks: 156 Math Focus: Abstract algebra 
It is not true. For example $A_4$ has distinct Sylow 3subgroups, e.g. $P=\{e, (123), (132)\}$ and $P'=\{e, (124), (142)\}$, but $A_4=12$ so no subgroup of $A_4$ can have an order divisible by $9$.

June 4th, 2014, 10:57 AM  #3 
Senior Member Joined: Mar 2012 Posts: 294 Thanks: 88 
What the OP was probably trying to prove is something like this: Suppose that $P = P' = 9$, and that $P \cap P' = 3$. Then: $9\mid N_G(P \cap P')$. In this (special) case, we have that $P,P'$ are abelian, so that: $P \subseteq N(P \cap P')$, since any subgroup of $P$ (or $P'$) is normal in $P$ (respectively, $P'$). The divisibility conclusion then follows by Lagrange. In fact, we can generalize Olinguito's counterexample to any group whose Sylow 3subgroups have order 3. 

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abstract, algebra, subgroup, sylow 
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