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If $G$ is a finite group, $H\leq G$ a subgroup, and $n$ divides $|G|$, then $n$ divides $[G:N_{G}(H)]$ times the number of elements $g\in G$ for which $|\langle H,g\rangle|$ is a divisor of $n$.

This appears as Theorem 6.1 of van der Lee - Elementary proof of a theorem of Hawkes, Isaacs, and Özaydin. I was not able to find a source myself, and I wondered whether the result was new. Any references for this result?

(When $H=1$, this is just Frobenius' theorem, by which the number of $g\in G$ with $g^{n}=1$ is a multiple of $n$, for $n\mid |G|$.)

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  • $\begingroup$ arxiv.org/abs/1907.00513 $\endgroup$ Jul 3, 2019 at 14:52
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    $\begingroup$ it would have been better to put the link to arxiv paper already in the question... $\endgroup$ Jul 3, 2019 at 20:03
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    $\begingroup$ I have edited in your remarks from the comments. Please feel free to revert or correct if you like. $\endgroup$
    – LSpice
    Jul 4, 2019 at 20:32
  • $\begingroup$ I apologise for the edit; I tried only to incorporate the information from the comments. Of course, please feel free to revert or correct if the edit is not welcome. $\endgroup$
    – LSpice
    Jul 4, 2019 at 23:25
  • $\begingroup$ @LSpice, sorry, no problem. $\endgroup$ Jul 5, 2019 at 8:29

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