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While reading through crazyproject, I came across the following proof that there were no simple groups of order $9555$. However, I do not understand the step that says:

"Moreover, since 7 does not divide 12 and 13 does not divide 48, $P_7P_{13}$ is abelian."

I don't quite follow this reasoning. If someone could explain this to me I'd be very grateful.

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    $\begingroup$ Are you familiar with the Sylow Subgroup Theorems? $\endgroup$
    – graydad
    Oct 23, 2014 at 1:01
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    $\begingroup$ The proof should follow from them... $\endgroup$
    – graydad
    Oct 23, 2014 at 1:03
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    $\begingroup$ Give me a sec to type mate $\endgroup$
    – graydad
    Oct 23, 2014 at 1:14
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    $\begingroup$ It is not crazy hard but I wanted to review my algebra before I give you a definitive answer. I haven't done this stuff in a while, although I do know this type of proof follows directly from the Sylow Theorems. I am trying to help you but I do not appreciate the insistence with which I am being asked to answer the question. If you really want to know so bad, I'm sure you can google it faster than I can get back to you. Or you can wait for another user to answer. $\endgroup$
    – graydad
    Oct 23, 2014 at 1:28
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    $\begingroup$ @graydad I apologize. Please do take your time. :) $\endgroup$
    – user151882
    Oct 23, 2014 at 1:29

1 Answer 1

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The idea is to show that elements of $P_7$ and $P_{13}$ commute. The proof seems to be using that $|\mathrm{Aut}(P_7)| = 48$ but this is wrong. Because $|P_7| = 49$ we know that $P_7$ isomorphic to one of $\mathbb{Z}_{49}$ or $\mathbb{Z}_{7} \times \mathbb{Z}_{7}$ and so $|\mathrm{Aut}(P_7)|$ is either $42$ or $48 \cdot 42$.

The idea of the proof can still be used. We know that $P_{13} \leq N_G(P_7)$ and $13$ does not divide $|\mathrm{Aut}(P_7)|$ so we must have $P_{13} \leq C_G(P_7)$. Therefore $P_7P_{13}$ is abelian, since both $P_7$ and $P_{13}$ are.

We could change the proof to avoid thinking about $P_7P_{13}$ as follows. Since $P_{13} \leq C_G(P_7)$ we also have that $P_7 \leq C_G(P_{13}) \leq N_G(P_{13})$ and then pick up the proof in the last sentence.

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