transfinite derived series

The transfinite derived series of a group is an extension of its derived series, defined as follows. Let $G$ be a group and let $G^{(0)}=G$. For each ordinal $\alpha$ let $G^{(\alpha +1)}$ be the derived subgroup of $G^{(\alpha )}$. For each limit ordinal $\delta$ let $G^{(\delta )}={\bigcap }_{\alpha \in \delta }{G}^{(\alpha )}$.

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 $\alpha$ such that $G^{(\alpha +1)}=G^{(\alpha )}$. All remaining terms of the series are then equal to $G^{(\alpha )}$, 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 subgroup (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 quotient (http://planetmath.org/QuotientGroup) $G/𝒫G$ is hypoabelian, and is sometimes called the hypoabelianization of $G$ (by analogy with the abelianization).

A group $G$ for which $G^{(n)}$ is trivial for some finite $n$ is called a solvable group. A group $G$ for which $G^{(\omega )}$ (the intersection of the derived series) is trivial is called a residually solvable group. Free groups (http://planetmath.org/FreeGroup) 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