transfinite derived series

The transfinite derived series of a group is an extensionPlanetmathPlanetmathPlanetmath of its derived series, defined as follows. Let G be a group and let G(0)=G. For each ordinalMathworldPlanetmathPlanetmath α let G(α+1) be the derived subgroup of G(α). For each limit ordinalMathworldPlanetmath δ let G(δ)=αδG(α).

Every member of the transfinite derived series of G is a fully invariant subgroup of G.

The transfinite derived series eventually terminates, that is, there is some ordinal α such that G(α+1)=G(α). All remaining terms of the series are then equal to G(α), which is called the perfect radical or maximum perfect subgroup of G, and is denoted 𝒫G. As the name suggests, 𝒫G is perfect, and every perfect subgroupMathworldPlanetmathPlanetmath ( of G is a subgroup of 𝒫G. A group in which the perfect radical is trivial (that is, a group without any non-trivial perfect subgroups) is called a hypoabelian group. For any group G, the quotientPlanetmathPlanetmath ( G/𝒫G is hypoabelian, and is sometimes called the hypoabelianization of G (by analogyMathworldPlanetmath with the abelianizationMathworldPlanetmath).

A group G for which G(n) is trivial for some finite n is called a solvable groupMathworldPlanetmath. A group G for which G(ω) (the intersectionMathworldPlanetmath of the derived series) is trivial is called a residually solvable group. Free groupsMathworldPlanetmath ( of rank greater than 1 are examples of residually solvable groups that are not solvable.

Title transfinite derived series
Canonical name TransfiniteDerivedSeries
Date of creation 2013-03-22 14:16:33
Last modified on 2013-03-22 14:16:33
Owner yark (2760)
Last modified by yark (2760)
Numerical id 14
Author yark (2760)
Entry type Definition
Classification msc 20F19
Classification msc 20F14
Related topic DerivedSubgroup
Defines perfect radical
Defines maximum perfect subgroup
Defines hypoabelianization
Defines hypoabelianisation