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

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- Oct 31st 2009, 09:31 PMmakenqau1group 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 PMtonio

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 AMSwlabr