If the order of a group is prime number then the group is cyclic?

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?

Re: If the order of a group is prime number then the group is cyclic?

Don't worry I found the solution. The answer is on Pinter's Abstract Algebra book page-111:

*If is any group and , it is easy to see that:*

.......

c) the set of all the powers of is a subgroup of

This subgroup is called the cyclic subgroup of generated by .

So the answer to my problem in my last post:

That's it. That's all I wanted to know.

Re: If the order of a group is prime number then the group is cyclic?

to answer your earlier question as asked:

no, for any element a of G, any power of a is ALSO in G, by closure.

Re: If the order of a group is prime number then the group is cyclic?

Quote:

Originally Posted by

**Deveno** to answer your earlier question as asked:

no, for any element a of G, any power of a is ALSO in G, by closure.

Thanks Deveno.