- #1
jumpr
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Homework Statement
Prove that any subgroup of a finitely generated abelian group is finitely generated.
Homework Equations
The Attempt at a Solution
I've attempted a proof by induction on the number of generators. The case n=1 corresponds to a cyclic group, and any subgroup of a cyclic group is cyclic, and so generated by one element. Then for the inductive step, I supposed that G was finitely generated, say [itex]G = \mathbb{Z} a_{1} + ... + \mathbb{Z} a_{n}[/itex], and that the result holds for groups generated by fewer than n elements. I've then let [itex]H \le G[/itex], and considered the quotient group [itex]G/\mathbb{Z}a_{n}[/itex], and then hoped that the correspondence theorem would help me out, but so far I can't seem to make it work.
Am I even attacking this problem correctly?