In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent. In other words, it has a normal nilpotent subgroup such that the quotient group is also nilpotent.
In symbols, is metanilpotent if there is a normal subgroup such that both and are nilpotent.
The following are clear:
Every metanilpotent group is a solvable group. Every subgroup and every quotient of a metanilpotent group is metanilpotent.
References J.C. Lennox, D.J.S. Robinson, The Theory of Infinite Soluble Groups, Oxford University Press, 2004, . P.27. D.J.S. Robinson, A Course in the Theory of Groups, GTM 80, Springer Verlag, 1996, . P.150.