metabelian group
Definition
A metabelian group^{} is a group $G$ that possesses a normal subgroup^{} $N$ such that $N$ and $G/N$ are both abelian^{}. Equivalently, $G$ is metabelian if and only if the commutator subgroup^{} $[G,G]$ is abelian. Equivalently again, $G$ is metabelian if and only if it is solvable of length at most $2$.
(Note that in older literature the term tends to be used in the stronger sense that the central quotient $G/Z(G)$ is abelian. This is equivalent^{} to being nilpotent^{} of class at most $2$. We shall not use this sense here.)
Examples

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All abelian groups.
 •

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All groups of order less than $24$.

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All metacyclic groups^{}.
Properties
Subgroups^{} (http://planetmath.org/Subgroup), quotients (http://planetmath.org/QuotientGroup) and (unrestricted) direct products^{} of metabelian groups are also metabelian. In other words, metabelian groups form a variety^{} (http://planetmath.org/VarietyOfGroups); they are, in fact, the groups in which $({w}^{1}{x}^{1}wx)({y}^{1}{z}^{1}yz)=({y}^{1}{z}^{1}yz)({w}^{1}{x}^{1}wx)$ for all elements $w$, $x$, $y$ and $z$.
Title  metabelian group 

Canonical name  MetabelianGroup 
Date of creation  20130322 15:36:42 
Last modified on  20130322 15:36:42 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  8 
Author  yark (2760) 
Entry type  Definition 
Classification  msc 20E10 
Classification  msc 20F16 
Synonym  metaabelian group 
Related topic  AbelianGroup2 
Defines  metabelian 
Defines  metaabelian 