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When finding the order of a cyclic group, do we determine so by counting the number of elements in that group generator by the cyclic group?

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    $\begingroup$ The order of any group is the number of elements in the group. $\endgroup$ – Cameron Williams Jun 13 '15 at 4:47
  • $\begingroup$ For any finite group (cyclic or otherwise) the order of the group is the number of elements in that group. $\endgroup$ – Anurag A Jun 13 '15 at 4:47
  • $\begingroup$ @AnuragA the group does not have to be finite. $\endgroup$ – Mr.Fry Jun 13 '15 at 4:51
  • $\begingroup$ @Mr.Fry the reason I added finite was because OP was using the term "counting the number" of elements which would make more sense in a finite setting. $\endgroup$ – Anurag A Jun 13 '15 at 4:54
  • $\begingroup$ There are sets which are countably infinite. So counting make sense outside of the finite case. There would be confusion for the OP seeing your reply and Cameron's. $\endgroup$ – Mr.Fry Jun 13 '15 at 4:56
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The order of any group is the number of elements in the group. Same holds for cyclic groups. Since moreover cyclic groups have a generator say $x$. Then the order of this cyclic group will be the order of $x$.

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