# Math Help - Prove a Group is NOT cyclic

1. ## Prove a Group is NOT cyclic

How would I show that a group of positve rational numbers under multiplication is not cyclic.

I can see that its true, but how would I prove it

2. Originally Posted by Ranger SVO
How would I show that a group of positve rational numbers under multiplication is not cyclic.

I can see that its true, but how would I prove it
Perhaps proof by contradiction. Assume there's an element g that generates all the other elements.

3. Originally Posted by Ranger SVO
How would I show that a group of positve rational numbers under multiplication is not cyclic.

I can see that its true, but how would I prove it
Suppose it is. Let $x$ be the generator. If $x$ is an integer then $x^n$ is an integer which means it cannot generate the non-integers. If $x$ is a non-integer then $x^n$ is a non-integer which means it cannot generate the integers.