# Fitting length > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Fitting_length > Markdown URL: https://mediated.wiki/source/Fitting_length.md > Source: https://en.wikipedia.org/wiki/Fitting_length > Source revision: 1029479408 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [mathematics](/source/Mathematics), specifically in the area of [algebra](/source/Algebra) known as [group theory](/source/Group_theory), the **Fitting length** (or **nilpotent length**) measures how far a [solvable group](/source/Solvable_group) is from being [nilpotent](/source/Nilpotent_group). The concept is named after [Hans Fitting](/source/Hans_Fitting), due to his investigations of nilpotent [normal subgroups](/source/Normal_subgroups). ## Definition A **Fitting chain'*(or****Fitting series**or '***) for a [group](/source/Group_(mathematics)) is a [subnormal series](/source/Subnormal_series) with [nilpotent](/source/Nilpotent_group) [quotients](/source/Quotient_group). In other words, a finite sequence of [subgroups](/source/Subgroup) including both the whole group and the trivial group, such that each is a [normal subgroup](/source/Normal_subgroup) of the previous one, and such that the quotients of successive terms are nilpotent groups. The **Fitting length** or **nilpotent length** of a [group](/source/Group_(mathematics)) is defined to be the smallest possible length of a Fitting chain, if one exists. ## Upper and lower Fitting series Just as the [upper central series](/source/Upper_central_series) and [lower central series](/source/Lower_central_series) are extremal among [central series](/source/Central_series), there are analogous series extremal among nilpotent series. For a finite group *H*, the [Fitting subgroup](/source/Fitting_subgroup) *Fit*(*H*) is the maximal normal nilpotent subgroup, while the minimal normal subgroup such that the quotient by it is nilpotent is *γ*∞(*H*), the intersection of the (finite) [lower central series](/source/Lower_central_series), which is called the [nilpotent residual](/source/Nilpotent_residual). These correspond to the center and the commutator subgroup (for upper and lower central series, respectively). These do not hold for infinite groups, so for the sequel, assume all groups to be finite. The **upper Fitting series** of a finite group is the sequence of characteristic subgroups *Fit**n*(*G*) defined by *Fit*0(*G*) = 1, and *Fit**n*+1(*G*)/*Fit**n*(*G*) = *Fit*(G/*Fit**n*(*G*)). It is an ascending nilpotent series, at each step taking the *maximal* possible subgroup. The **lower Fitting series** of a finite group *G* is the sequence of [characteristic subgroups](/source/Characteristic_subgroup) *F**n*(*G*) defined by *F*0(*G*) = *G*, and *F**n*+1(*G*) = *γ*∞(*F**n*(*G*)). It is a descending nilpotent series, at each step taking the *minimal* possible subgroup. ## Examples - A nontrivial group has Fitting length 1 if and only if it is nilpotent. - The [symmetric group on three points](/source/Dihedral_group_of_order_6) has Fitting length 2. - The [symmetric group on four points](/source/Tetrahedral_group) has Fitting length 3. - The [symmetric group](/source/Symmetric_group) on five or more points has no Fitting chain at all, not being solvable. - The iterated wreath product of *n* copies of the symmetric group on three points has Fitting length 2*n*. ## Properties - A group has a Fitting chain if and only if it is [solvable](/source/Solvable_group). - The lower Fitting series is a Fitting chain if and only if it eventually reaches the trivial subgroup, if and only if *G* is solvable. - The upper Fitting series is a Fitting chain if and only if it eventually reaches the whole group, *G*, if and only if *G* is solvable. - The lower Fitting series descends most quickly amongst all Fitting chains, and the upper Fitting series ascends most quickly amongst all Fitting chains. Explicitly: For every Fitting chain, 1 = *H*0 ⊲ *H*1 ⊲ … ⊲ *H**n* = *G*, one has that *H**i* ≤ *Fit**i*(*G*), and *F**i*(*G*) ≤ *H**n*−*i*. - For a solvable group, the length of the lower Fitting series is equal to length of the upper Fitting series, and this common length is the Fitting length of the group. More information can be found in (Huppert 1967, pp. Kap. III, §4). ## Connection between central series and Fitting series What [central series](/source/Central_series) do for nilpotent groups, Fitting series do for solvable groups. A group has a central series if and only if it is nilpotent, and a Fitting series if and only if it is solvable. Given a solvable group, the lower Fitting series is a "coarser" division than the lower central series: the lower Fitting series gives a series for the whole group, while the lower central series descends only from the whole group to the first term of the Fitting series. The lower Fitting series proceeds: - *G* = *F*0 ⊵ *F*1 ⊵ ⋯ ⊵ 1, while the lower central series subdivides the first step, - *G* = *G*1 ⊵ *G*2 ⊵ ⋯ ⊵ *F*1, and is a lift of the lower central series for the first quotient *F*0/*F*1, which is nilpotent. Proceeding in this way (lifting the lower central series for each quotient of the Fitting series) yields a subnormal series: - *G* = *G*1 ⊵ *G*2 ⊵ ⋯ ⊵ *F*1 = *F*1,1 ⊵ *F*1,2 ⊵ ⋯ ⊵ *F*2 = *F*2,1 ⊵ ⋯ ⊵ *F*n = 1, like the coarse and fine divisions on a [ruler](/source/Ruler). The successive quotients are abelian, showing the equivalence between being solvable and having a Fitting series. ## See also - [Central series](/source/Central_series) - [3-step group](/source/3-step_group) ## References - Huppert, B. (1967), *Endliche Gruppen* (in German), Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), ISBN 978-3-540-03825-2. MR 0224703. [OCLC 527050](https://www.worldcat.org/oclc/527050) --- Adapted from the Wikipedia article [Fitting length](https://en.wikipedia.org/wiki/Fitting_length) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Fitting_length?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.