Upper Fitting series
The upper Fitting series of a finite group is an ascending subgroup series defined as follows:
- The zeroth member, , is the trivial subgroup.
- Each member is defined so that the quotient group is the Fitting subgroup of . The Fitting subgroup of a group is defined as the join of all nilpotent normal subgroups. In particular, for a finite group, the Fitting subgroup itself is a nilpotent normal subgroup.
The series reaches the whole group if and only if the group is solvable, i.e., is a finite solvable group. The upper Fitting series of a finite solvable group is the fastest ascending Fitting series and hence its length equals the Fitting length.
In general, for a finite possibly non-solvable group, the upper Fitting series terminates (or stabilizes) at the solvable radical, which is the unique largest solvable normal subgroup. The quotient by the solvable radical is a Fitting-free group.