# Group Question

1. Jul 20, 2008

### futurebird

From "The Theory of Groups" by Rotman

2.5. Prove that the multiplicative group of positive rationals is generated by all rationals of the form:
$$\frac{1}{p}$$,
where p is prime. ​

... um... no it's not. Right? How can I prove this when I don't even think it is true? I mean, for example take the positive rational number 75. How can I generate that using $$\frac{1}{p^{n}}$$, where p is prime, $$n \in \mathbb{N}$$? The largest possible value for $$\frac{1}{p^{n}}$$ is 1/2...

I could see how this would be possible if we had addition as the operation for the generating set... but, then why is it the multiplicative group of positive rationals?

2. Jul 20, 2008

### Hurkyl

Staff Emeritus
You can't, but why would you restrict yourself to positive exponents?

3. Jul 20, 2008

Thanks!