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"A finite group is cyclic if, for any integer m, the number of elements of order dividing m is at most m."

I have never seen this characterisation of cyclic groups before. How do I prove this? I hope the proof isn't too complicated because it's a lemma in another proof.

John Michael
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  • This is usually part of the proof that any finite subgroup of the multiplicative group of a field is cyclic, but it stands on its own, though you may argue it's not too natural. See http://math.stackexchange.com/a/59911/589. – lhf May 15 '13 at 11:36
  • Someone has also asked the question without the field theory context here. – Karl Kroningfeld May 15 '13 at 11:43

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