# Fitting length

﻿
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.

## Definition

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.

## Upper and lower Fitting series

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 normal 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.

The upper Fitting series of a finite group is the sequence of characteristic subgroups Fitn(G) defined by Fit0(G) = 1, and Fitn+1(G)/Fitn(G) = Fit(G/Fitn(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 Fn(G) defined by F0(G) = G, and Fn+1(G) = γ(Fn(G)). It is a descending nilpotent series, at each step taking the minimal possible subgroup.

## Examples

• A group has Fitting length 1 if and only if it is nilpotent.
• The symmetric group on three points has Fitting length 2.
• The symmetric group on four points has Fitting length 3.
• The 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 2n.

## Properties

• A group has a Fitting chain if and only if it is solvable.
• 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 = H0H1 ⊲ … ⊲ Hn = G, one has that HiFiti(G), and Fi(G) ≤ Hni.
• 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.

## Connection between central series and Fitting series

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.

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 = F0F1 ⊵ ⋯ ⊵ 1,

while the lower central series subdivides the first step,

G = G1G2 ⊵ ⋯ ⊵ 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 = G1G2 ⊵ ⋯ ⊵ F1 = F1,1F1,2 ⊵ ⋯ ⊵ 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.

## References

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Fitting — can refer to: # Any machine, piping or tubing part that can attach or connect two or more larger parts. For examples, see coupling, compression fitting or piping and plumbing fittings. # The process of applying regression analysis to data. This… …   Wikipedia

• Fitting subgroup — In mathematics, especially in the area of algebra known as group theory, the Fitting subgroup F of a finite group G , named after Hans Fitting, is the unique largest normal nilpotent subgroup of G . Intuitively, it represents the smallest… …   Wikipedia

• length at first maturity — length at which 50% of the individuals of a given sex are considered to be reproductively mature. Usually based on females and estimated by fitting a logistic curve to the relationships between proportion mature and length. Abbreviated as L50.… …   Dictionary of ichthyology

• Fitting lemma — The Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in abstract algebra. Suppose M is a module over some ring. If M is indecomposable and has finite length, then every endomorphism of M is either bijective or… …   Wikipedia

• Channel length modulation — Cross section of a MOSFET operating in the saturation region One of several short channel effects in MOSFET scaling, channel length modulation (CLM) is a shortening of the length of the inverted channel region with increase in drain bias for… …   Wikipedia

• Minimum message length — (MML) is a formal information theory restatement of Occam s Razor: even when models are not equal in goodness of fit accuracy to the observed data, the one generating the shortest overall message is more likely to be correct (where the message… …   Wikipedia

• Minimum description length — The minimum description length (MDL) principle is a formalization of Occam s Razor in which the best hypothesis for a given set of data is the one that leads to the best compression of the data. MDL was introduced by Jorma Rissanen in 1978. It is …   Wikipedia

• Audio tape length and thickness — Since the widespread adoption of reel to reel audio tape recording in the 1950s, audio tapes and tape cassettes have been available in many formats. This article describes the length, tape thickness and playing times of some of the most common… …   Wikipedia

• fractional fitting — /frækʃənəl ˈfɪtɪŋ/ (say frakshuhnuhl fiting) noun a sizing system for shoes, providing a number of variations in width in each length size …   Australian English dictionary

• Clerical Costume —     Clerical Costume     † Catholic Encyclopedia ► Clerical Costume     To discuss the question of ecclesiastical costume in any detail would be impossible in an article like the present. No topic has formed the subject of so many synodal… …   Catholic encyclopedia