hope you all doing fine. I have a question. Is it true that a abelian-by-(finite abelian) group is also (finite abelian)-by-abelian? Thanks.
closed as off-topic by Travis, 6005, Ken, Harish Chandra Rajpoot, Michael Galuza Aug 26 '15 at 5:04
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In fact it is not true in either direction.
The infinite dihedral group is abelian-by-finite abelian, but not (finite abelian)-by-abelian.
A central product of infinite many copies of $D_8$ is (finite abelian)-by-abelian, but not abelian-by-(finite abelian).