# Thread: 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 $\displaystyle x$ be the generator. If $\displaystyle x$ is an integer then $\displaystyle x^n$ is an integer which means it cannot generate the non-integers. If $\displaystyle x$ is a non-integer then $\displaystyle x^n$ is a non-integer which means it cannot generate the integers.

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# Show that the group of positive rational numbers under multiplica- tion is not cyclic.

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