Fitting length
Encyclopedia
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.
is a subnormal series with nilpotent
quotients
. In other words, a finite sequence of 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 group
is defined to be the smallest possible length of a Fitting chain, if one exists.
, 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 subgroup
s 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.
More information can be found in .
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:
while the lower central series subdivides the first step,
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:
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in the area of algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
known as group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the Fitting length (or nilpotent length) measures how far a solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
is from being nilpotent
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
. The concept is named after Hans Fitting
Hans Fitting
Hans Fitting was a mathematician who worked in group theory...
, due to his investigations of nilpotent normal subgroups.
Definition
A Fitting chain (or Fitting series or ) for a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is a subnormal series with nilpotent
Nilpotent group
In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
quotients
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
. In other words, a finite sequence of subgroup
Subgroup
In group theory, 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 of G if the restriction of * to H x H is a group operation on H...
s including both the whole group and the trivial group, such that each is a normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of the previous one, and such that the quotients of successive terms are nilpotent groups.
The Fitting length or nilpotent length of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
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 seriesCentral 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...
, there are analogous series extremal among nilpotent series.
For a finite group H, the Fitting subgroup
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 subgroup which "controls" the structure of G when G is solvable...
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 subgroup
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...
s 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 pointsDihedral group of order 6The smallest non-abelian group has 6 elements. It is a dihedral group with notation D3 and the symmetric group of degree 3, with notation S3....
has Fitting length 2. - The symmetric group on four points has Fitting length 3.
- The symmetric groupSymmetric groupIn mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
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 solvableSolvable groupIn mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
. - 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 = H0 ⊲ H1 ⊲ … ⊲ Hn = G, one has that Hi ≤ Fiti(G), and Fi(G) ≤ Hn−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
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...
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
Ruler
A ruler, sometimes called a rule or line gauge, is an instrument used in geometry, technical drawing, printing and engineering/building to measure distances and/or to rule straight lines...
.
The successive quotients are abelian, showing the equivalence between being solvable and having a Fitting series.