page_title: Fitting length
section_title: Connection between central series and Fitting series
hierarchical_section_title: Fitting length / Connection between central series and Fitting series
caption_attribution_description: English: Closeup of a steel ruler
context_page_description: In mathematics, especially in the area of algebra known as group theory, the Fitting length measures how far a solvable group is from being nilpotent. The concept is named after Hans Fitting, due to his investigations of nilpotent normal subgroups.
context_section_description: What 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 = F₀ ⊵ F₁ ⊵ ⋯ ⊵ 1,
  while the lower central series subdivides the first step,
  G = G₁ ⊵ G₂ ⊵ ⋯ ⊵ F₁,
  and is a lift of the lower central series for the first quotient F₀/F₁, which is nilpotent.
  Proceeding in this way (lifting the lower central series for each quotient of the Fitting series) yields a subnormal series:
  G = G₁ ⊵ G₂ ⊵ ⋯ ⊵ F₁ = F1,1 ⊵ F1,2 ⊵ ⋯ ⊵ F₂ = F2,1 ⊵ ⋯ ⊵ Fₙ = 1,
  like the coarse and fine divisions on a ruler.
  The successive quotients are abelian, showing the equivalence between being solvable and having a Fitting series.
caption: Combining the lower Fitting series and lower central series on a solvable group yields a series with coarse and fine divisions, like the coarse and fine marks on a ruler.
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