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 generalized dihedral 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 $z$ .

单词	MetabelianGroup
释义	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 $z$ .
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