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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 3-Sylow 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 3-subgroups, 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 counter-example to any group whose Sylow 3-subgroups have order 3. Thanks from Olinguito

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