Fitting length

In mathematics, especially in the area of algebra known as group theory, the Fitting length (or nilpotent 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.

A Fitting chain (or Fitting series or nilpotent series) for a group is a subnormal series with nilpotent quotients. In other words, a finite sequence of subgroups including both the whole group and the trivial group, such that each is a 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 is defined to be the smallest possible length of a Fitting chain, if one exists.

Just as the upper central series and lower central series are extremal among central series, there are analogous series extremal among nilpotent series.

For a finite group H, the Fitting subgroup Fit(H) is the maximal normal nilpotent subgroup, while the minimal subgroup such that the quotient by it is nilpotent is γ_∞(H), the intersection of the (finite) lower central series, which is called the 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.

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