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 $\P{G}$ As the name suggests, $\P{G}$ is perfect, and every perfect subgroup of $G$ is a subgroup of $\P{G}$ 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 $G/\P{G}$ 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 of rank greater than $1$ are examples of residually solvable groups that are not solvable.