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Difference between revisions of "Meta-Abelian group"

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''metabelian group''
 
''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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063490/m0634901.png" /></td> </tr></table>
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$$[[x,y],[z,t]]=1.$$
  
 
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)
How to Cite This Entry:
Meta-Abelian group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meta-Abelian_group&oldid=13782
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article