Meta-Abelian Group

metabelian group

A solvable group of derived length two, i.e. a group whose commutator subgroup is Abelian. The family of all metabelian groups is a variety (see Variety of groups) defined by the identity

$$[[x,y],[z,t]]=1.$$

Special interest is attached to finitely-generated metabelian groups. These are all residually finite (see Residually-finite group) and satisfy the maximum condition (see Chain condition) for normal subgroups. An analogous property is shared by a generalization of these groups — the finitely-generated groups for which the quotient by an Abelian normal subgroup is polycyclic (see Polycyclic group).

In the Russian mathematical literature, by a metabelian group one sometimes means a nilpotent group of nilpotency class 2.

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