# 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.

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |

[2] | M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) |

#### Comments

#### References

[a1] | D.J.S. Robinson, "A course in the theory of groups" , Springer (1980) |

**How to Cite This Entry:**

Meta-Abelian group.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Meta-Abelian_group&oldid=31663