Upper Fitting series

Definition
The upper Fitting series of a finite group $$G$$ is an ascending subgroup series defined as follows:


 * The zeroth member, $$\operatorname{Fit}_0(G)$$, is the trivial subgroup.
 * Each member $$\operatorname{Fit}_i(G)$$ is defined so that the quotient group $$\operatorname{Fit}_i(G)/\operatorname{Fit}_{i-1}(G)$$ is the Fitting subgroup of $$G/\operatorname{Fit}_{i-1}(G)$$. 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 defining ingredient::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.

Related notions

 * Upper central series
 * Lower Fitting series
 * Derived series