Upper Fitting series: Difference between revisions
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{{quotient-iterated series|Fitting subgroup}} | {{quotient-iterated series|Fitting subgroup}} | ||
==Definition== | |||
The '''upper Fitting series''' of a [[finite group]] <math>G</math> is an ascending subgroup series defined as follows: | |||
* The zeroth member, <math>\operatorname{Fit}_0(G)</math>, is the trivial subgroup. | |||
* Each member <math>\operatorname{Fit}_i(G)</math> is defined so that the [[quotient group]] <math>\operatorname{Fit}_i(G)/\operatorname{Fit}_{i-1}(G)</math> is the [[Fitting subgroup]] of <math>G/\operatorname{Fit}_{i-1}(G)</math>. The [[Fitting subgroup]] of a group is defined as the [[join of subgroups|join]] of all [[nilpotent normal subgroup]]s. 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 group|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]] | |||
Latest revision as of 14:58, 2 August 2011
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 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.