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May 27th, 2014, 12:31 AM   #1
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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.

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Last edited by skipjack; June 4th, 2014 at 08:00 AM.
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June 3rd, 2014, 03:46 PM   #2
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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 $9$.
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June 4th, 2014, 10:57 AM   #3
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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.
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