A group 
such that the quotient group 
, where 
is the group center of
, is Abelian.
An equivalent condition is that the commutator
subgroup 
is contained in 
.
Metabelian Group
This entry contributed by Margherita Barile
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Barile, Margherita. "Metabelian Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/MetabelianGroup.html