Let be a group with a prime number of elements. If where ,
then the order of is some integer .
But then the cyclic group has elements.
By Lagrange's theorem, must be a factor of .
But is a prime number, and therefore .
It follows that has elements, and is therefore all of !
If is a group with a prime number of elements, then is a cyclic group. Furthermore,
any element in is a generator of .
The author said that:
"But then the cyclic group has elements."
I understand that the cyclic group has elements. No problem there. I understand this statement is true for the generator .
But why the group has to be definitely cyclic at the end?
Why the cyclic group ?
Can't there be situations where because we didn't say that is a subgroup of ?
So how did he come to the conclusion that is cyclic?