mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, specifically in the area of

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

known as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the Fitting length (or nilpotent length) measures how far a

solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates ...

is from being

nilpotent In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the class ...

. The concept is named after Hans Fitting, due to his investigations of nilpotent normal subgroups.

Definition

A Fitting chain (or Fitting series or ) for a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

is a

subnormal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups: :1 = A_0 \leq A_1 \leq \cdots \leq A_n = G where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simple ...

with

quotients. In other words, a finite sequence of

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

s 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

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 In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central ...

and

lower central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a centra ...

are extremal among

central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central ...

, there are analogous series extremal among nilpotent series. For a finite group ''H'', the

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 smalles ...

''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)

, 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 ''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 subgroup In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphi ...

s ''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 has Fitting length 2. * The symmetric group on four points has Fitting length 3. * The

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...

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. * 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 .

Connection between central series and Fitting series

What

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 A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...

. The successive quotients are abelian, showing the equivalence between being solvable and having a Fitting series.

References