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Let p be an odd prime number, and let n be a positive integer divisible by p. Show that $D_n$ has only one p-Sylow subgroup

I'm trying to prove this by saying np($D_n$) = ap+1 and a must be 0, but I'm stuck at the process of proving a have to be 0. How can I achieve it, thank you very much! (np(group G) means the number of p-Sylow subgroup in the group G)

It's solved, thanks to Arturo Magidin!

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  • $\begingroup$ Welcome to MSE. Please include your question in the body of the question, instead of putting it only in the title. $\endgroup$ – José Carlos Santos Dec 3 '19 at 6:37
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    $\begingroup$ It’s simpler than that: show that $D_n$ has a normal cyclic group of order $n$. Then show that this cyclic normal subgroup contains a Sylow $p$-subgroup of $G$, and deduce that it is normal. Then use the Second Sylow Theorem to conclude it is the only Sylow $p$-subgroup of $G$. $\endgroup$ – Arturo Magidin Dec 3 '19 at 6:55