>> (For example, if you have a group of size X, it has subgroups of every 
>> size that is a factor of X - including 1 and X itself.
> 
> I assume by "group" you actually mean to say "cyclic group"?  If not 
> that might explain why it's proving hard to understand.

I was under the impression the result applies to all groups - which is 
why the groups of prime order are always cyclic.

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