Cyclic Groups

Date: 02/27/2003 at 20:07:48
From: Jenn
Subject: Cyclic Groups

Prove that the group of nonzero rational numbers under multiplication 
is not cyclic.

Date: 02/28/2003 at 03:33:08
From: Doctor Jacques
Subject: Re: Cyclic Groups

Hi Jenn,

If the group in question is cyclic, it is generated by a single 
element g, i.e. any element of the group is some power of g or its 
inverse.

As g is rational, we can write it as a fraction:

  g = p/q

where p and q are integers, and we may assume that they are 
relatively prime, i.e. that the fraction is in lowest terms.

All powers of g and its inverse are of the form

  g^k = (p^k)/(q^k)
  (1/g)^k = (q^k)/(p^k)

Now, p and q are integers, so they have at most a finite set of prime 
factors, and it is possible to find a prime number r that divides 
neither p or q.

Can you show that r does not belong to the group generated by g?

Please feel free to write back if you want to discuss this further.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/