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 !

Conclusion:

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?