transfinite derived series
The transfinite derived series of a group isan extension of its derived series, defined as follows.Let be a group and let .For each ordinal
let be the derived subgroup of .For each limit ordinal
let .
Every member of the transfinite derived series of is a fully invariant subgroup of .
The transfinite derived series eventually terminates, that is,there is some ordinal such that .All remaining terms of the series are then equal to ,which is called the perfect radical or maximum perfect subgroupof , and is denoted .As the name suggests, is perfect,and every perfect subgroup (http://planetmath.org/Subgroup) of is a subgroup of .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 , the quotient
(http://planetmath.org/QuotientGroup) is hypoabelian, and is sometimes called the hypoabelianization of (by analogy
with the abelianization
).
A group for which is trivial for some finite is called a solvable group.A group for which (the intersection
of the derived series)is trivial is called a residually solvable group.Free groups
(http://planetmath.org/FreeGroup) of rank greater than 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 |