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

• All abelian groups.

• All generalized dihedral groups.

• All groups of order less than $24$.

• 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 $z$.