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The number of elements of order $n$ in a finite cyclic group of order $N$ is $0$ unless $n|N$, in which case it is $N/n$.

Is "the number of elements of order $n$" referring to the number of elements of the subgroup that is of order $n$?

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    No, it refers to the number of elements $x$ such that $x^n$ is the identity element, but $x^m$ is not for any $m.2012-08-18
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    You move just a bit too quick for me! I write my last answer to your other question, and you enhance the question while I'm writing it. You write me a comment (thank you) asking me about the second half, so I type up a response, and then you edit it out of the question into another! But I caught up!2012-08-18
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    The answer by mixedmath addresses the details, but in short the statement in the question is completely wrong (where did you get this from?). It should read: The number of elements of order $n$ in a finite cyclic group of order $N$ is $0$ unless $n \mid N$, in which case it is $\phi(n)$ ([Euler phi function](http://en.wikipedia.org/wiki/Euler_phi_function)).2012-08-18
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    @mixedmath: I know it is perfectly "legal" for someone to ask 4 questions their first day here, but surely something can be done to at least discourage people from doing as you describe in your comment? I mean, is anybody learning anything from this?2012-08-18

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