View source for Meta-Abelian group

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''metabelian group''

A [[Solvable group|solvable group]] of derived length two, i.e. a group whose [[Commutator subgroup|commutator subgroup]] is Abelian. The family of all metabelian groups is a variety (see [[Variety of groups|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|Residually-finite group]]) and satisfy the maximum condition (see [[Chain condition|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|Polycyclic group]]).

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer  (1979)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.J.S. Robinson,   "A course in the theory of groups" , Springer  (1980)</TD></TR></table>