Upper Fitting series

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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 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.

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