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 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 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 of rank greater than are examples of residually solvable groups that are not solvable.