# Abelian-by-(finite abelian) [closed]

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 GaluzaAug 26 '15 at 5:04

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• What do you mean by that? – Robert Israel Aug 25 '15 at 17:27
• You need to say what you mean by an A-by-B group, since there are two opposite conventions for this. – Derek Holt Aug 25 '15 at 17:29
• I meant that can we find a finite abelian normal subgroup N such that the quotient G/N is abelian? – Yalcin Aug 25 '15 at 17:30
• Oh I am sorry, I am new :) – Yalcin Aug 25 '15 at 17:30
• A-by-B means G has a normal subgroup N such that N has the property A, and the quotient G/N has the property B. – Yalcin Aug 25 '15 at 17:32

A central product of infinite many copies of $D_8$ is (finite abelian)-by-abelian, but not abelian-by-(finite abelian).