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单词 TransfiniteDerivedSeries
释义

transfinite derived series


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

Every member of the transfinite derived series of Gis 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 subgroupof G, and is denoted 𝒫G.As the name suggests, 𝒫G is perfect,and every perfect subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) 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 (http://planetmath.org/QuotientGroup) G/𝒫Gis 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 nis 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 (http://planetmath.org/FreeGroup) of rank greater than 1are examples of residually solvable groups that are not solvable.

Titletransfinite derived series
Canonical nameTransfiniteDerivedSeries
Date of creation2013-03-22 14:16:33
Last modified on2013-03-22 14:16:33
Owneryark (2760)
Last modified byyark (2760)
Numerical id14
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20F19
Classificationmsc 20F14
Related topicDerivedSubgroup
Definesperfect radical
Definesmaximum perfect subgroup
Defineshypoabelianization
Defineshypoabelianisation
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更新时间:2025/5/4 23:02:34