transfinite derived series
The transfinite derived series of a group is
an 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 subgroup
of , 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 |