Difference between revisions of "Meta-Abelian group"
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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 | 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]]). | 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]]). | ||
Latest revision as of 09:44, 13 April 2014
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) |
Meta-Abelian group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meta-Abelian_group&oldid=31663