# 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 $\mathcal{P}{G}$. As the name suggests, $\mathcal{P}{G}$ is perfect, and every perfect subgroup (http://planetmath.org/Subgroup) of $G$ is a subgroup of $\mathcal{P}{G}$. 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) $G/\mathcal{P}{G}$ 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