Upper Fitting series

This article defines a quotient-iterated series with respect to the following subgroup-defining function: Fitting subgroup

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 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