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I have the following question:

Let $G$ be a group and let $H$ be a subgroup of finite index of $G$. Let $|G:H|=n$ Then it holds: $g^{n!}\in H$ for all $g\in G$.

Why is this true?

I think, that's not very difficult, but I have no idea at the moment.

Thanks!

  • 1
    Have you tried the Pigeonhole principle?2012-02-13
  • 0
    Thank you for these nice solutions!2012-02-15
  • 3
    It seems you have asked many questions but have accepted answers to none of them. If you are satisfied with the answer of a question you have asked, you can accept it by clicking on the check mark below the up/down arrows. It lets other users know that this question has a good answer.2012-05-31
  • 0
    If $H$ is normal in $G$, we actually have $g^n \in H$ (see also https://math.stackexchange.com/questions/545417).2018-11-26
  • 0
    (See also https://math.stackexchange.com/questions/472672, if $H$ is normal in $G$).2018-11-28

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