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Abelian subgroups all finitely generated but an Abelian quotient group not finitely generated?
Does there exist a (non-Abelian) group G such that any Abelian subgroups are finitely generated, with a normal subgroup H, such that the quotient group G/H is Abelian, but not finitely generated.
Alternatively, given an Abelian group, A, which is not finitely generated (like the rational numbers under addition), and some other group H (pick any group you like), can a (semi-direct) product of A and H be constructed whose Abelian subgroups are all finitely generated?
(This is an attempt to answer Stoops' question about whether solvable groups with only finitely generated Abelian subgroups must be polycyclic.)
1 Answer
- Anonymous1 decade agoFavorite Answer
Take G to be a free group on infinitely many generators. Then the only abelian subgroups of G are isomorphic to Z, but G/[G,G] is free abelian on infinitely many generators.
This doesn't make G a semidirect product, though. In fact, your second question has the answer no, since a semidirect product of A and H always has A and H as subgroups. A would then be a not-finitely-generated abelian subgroup.