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Let $G$ be a group of order $2\cdot 17\cdot 23$. Let $P$ be a $17$-Sylow subgroup and $Q$ a $23$-Sylow subgroup. Show that $PQ$ is normal in $G$ and that $P$ is characteristic in $PQ$.

What I know: both $P$ and $Q$ are normal in $G$ by the Sylow theorems, $PQ$ is normal in $G$ because it has index 2.

So all that remains is to show that $P$ is characteristic in $PQ$. Now, certainly any inner automorphism of $PQ$ will fix $P$ since $17$-Sylow subgroups are all conjugate and $P$ is normal in $G$ (and hence unique). But how do I know there isn't some outer automorphism that pushes $P$ around?

More generally, it seems like this would be true for groups of order $2pq$ for primes $2<p<q$, since I doubt $17$ and $23$ are particularly significant. Thoughts?

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Hint. Use Sylow #3 to show that there's only one Sylow $17$-subgroup of $PQ$. Then show that a unique Sylow subgroup is characteristic.

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  • $\begingroup$ Your first comment is straight-forward. For your second, just to be safe: an automorphism $\sigma\in Aut(PQ)$ sends subgroups of PQ to subgroups of PQ, but automorphisms are bijective so $|\sigma(P)|=|P|$. By uniqueness, we then have $\sigma(P)=P$, so $P$ is characteristic in $PQ$. So by this we also have that $P$ and $Q$ are both characteristic in $G$, as well, correct? $\endgroup$ – Bey Nov 10 '13 at 23:47

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