# group of rational numbers

• Oct 31st 2009, 09:31 PM
makenqau1
group of rational numbers
How we can prove additive group of rational numbers (Q,+)is not cyclic, but every finitely generated subgroup of (Q,+)is cyclic.
• Oct 31st 2009, 10:46 PM
tonio
Quote:

Originally Posted by makenqau1
How we can prove additive group of rational numbers (Q,+)is not cyclic, but every finitely generated subgroup of (Q,+)is cyclic.

This question obviously belongs in Linear and Abstract Algebra and not here.

Assume Q = <a/b> and check the prime factorization of b. Now show that
any multiple m*(a/b) of a/b cannot have different primes in the denominator from the ones that appear in the decomposition of b and thus...

About every fin. gen. sbgp.: check first for a 2-generator group (pretty easy) and then generalize by induction

Tonio
• Nov 1st 2009, 01:48 AM
Swlabr
Quote:

Originally Posted by makenqau1
How we can prove additive group of rational numbers (Q,+)is not cyclic, but every finitely generated subgroup of (Q,+)is cyclic.

There is another nice property about this group - every (non-trivial) image is infinite, but every element of it has finite order. Can you prove this?