metabelian group


A metabelian groupMathworldPlanetmath is a group G that possesses a normal subgroupMathworldPlanetmath N such that N and G/N are both abelianMathworldPlanetmath. Equivalently, G is metabelian if and only if the commutator subgroupMathworldPlanetmath [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 equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to being nilpotentPlanetmathPlanetmathPlanetmath of class at most 2. We shall not use this sense here.)



SubgroupsMathworldPlanetmathPlanetmath (, quotients ( and (unrestricted) direct productsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of metabelian groups are also metabelian. In other words, metabelian groups form a varietyMathworldPlanetmathPlanetmath (; they are, in fact, the groups in which (w-1x-1wx)(y-1z-1yz)=(y-1z-1yz)(w-1x-1wx) for all elements w, x, y and z.

Title metabelian group
Canonical name MetabelianGroup
Date of creation 2013-03-22 15:36:42
Last modified on 2013-03-22 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 meta-abelian group
Related topic AbelianGroup2
Defines metabelian
Defines meta-abelian