HOME

TheInfoList



OR:

In
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 ...
, especially 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 subgroup ''F'' of a
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
''G'', named after Hans Fitting, is the unique largest
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
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 ...
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 ...
of ''G''. Intuitively, it represents the smallest subgroup which "controls" the structure of ''G'' when ''G'' is solvable. When ''G'' is not solvable, a similar role is played by the generalized Fitting subgroup ''F*'', which is generated by the Fitting subgroup and the components of ''G''. For an arbitrary (not necessarily finite) group ''G'', the Fitting subgroup is defined to be the subgroup generated by the nilpotent normal subgroups of ''G''. For infinite groups, the Fitting subgroup is not always nilpotent. The remainder of this article deals exclusively with
finite group Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...
s.


The Fitting subgroup

The nilpotency of the Fitting subgroup of a finite group is guaranteed by
Fitting's theorem Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows: :If ''M'' and ''N'' are nilpotent normal subgroups of a group ''G'', then their product ''MN'' is also a nilpotent normal subgroup of ''G''; if, mor ...
which says that the product of a finite collection of normal nilpotent subgroups of ''G'' is again a normal nilpotent subgroup. It may also be explicitly constructed as the product of the p-cores of ''G'' over all of the primes ''p'' dividing the order of ''G''. If ''G'' is a finite non-trivial solvable group then the Fitting subgroup is always non-trivial, i.e. if ''G''≠1 is finite solvable, then ''F''(''G'')≠1. Similarly the Fitting subgroup of ''G''/''F''(''G'') will be nontrivial if ''G'' is not itself nilpotent, giving rise to the concept of
Fitting length In mathematics, specifically 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 ni ...
. Since the Fitting subgroup of a finite solvable group contains its own
centralizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...
, this gives a method of understanding finite solvable groups as
extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * E ...
of nilpotent groups by faithful
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
s of nilpotent groups. In a nilpotent group, every
chief factor A factor is a type of trader who receives and sells goods on commission, called factorage. A factor is a mercantile fiduciary transacting business in his own name and not disclosing his principal. A factor differs from a commission merchant in ...
is centralized by every element. Relaxing the condition somewhat, and taking the subgroup of elements of a general finite group which centralize every chief factor, one simply gets the Fitting subgroup again : :\operatorname(G) = \bigcap \. The generalization to ''p''-nilpotent groups is similar.


The generalized Fitting subgroup

A component of a group is a subnormal
quasisimple In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension ''E'' of a simple group ''S''. In other words, there is a short exact sequence :1 \to Z(E) \to E \to S \to 1 such that E = , E/ ...
subgroup. (A group is quasisimple if it is a perfect central extension of a simple group.) The layer ''E''(''G'') or ''L''(''G'') of a group is the subgroup generated by all components. Any two components of a group commute, so the layer is a perfect central extension of a product of simple groups, and is the largest normal subgroup of ''G'' with this structure. The generalized Fitting subgroup ''F''*(''G'') is the subgroup generated by the layer and the Fitting subgroup. The layer commutes with the Fitting subgroup, so the generalized Fitting subgroup is a central extension of a product of ''p''-groups and
simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...
s. The layer is also the maximal normal semisimple subgroup, where a group is called semisimple if it is a perfect central extension of a product of simple groups. This definition of the generalized Fitting subgroup can be motivated by some of its intended uses. Consider the problem of trying to identify a normal subgroup ''H'' of ''G'' that contains its own centralizer and the Fitting group. If ''C'' is the centralizer of ''H'' we want to prove that ''C'' is contained in ''H''. If not, pick a minimal
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 ...
''M/Z(H)'' of ''C/Z(H)'', where ''Z(H)'' is the center of ''H'', which is the same as the intersection of ''C'' and ''H''. Then ''M''/''Z''(''H'') is a product of simple or
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s as it is characteristically simple. If ''M''/''Z''(''H'') is a product of cyclic groups then ''M'' must be in the Fitting subgroup. If ''M''/''Z''(''H'') is a product of non-abelian simple groups then the derived subgroup of ''M'' is a normal semisimple subgroup mapping onto ''M''/''Z''(''H''). So if ''H'' contains the Fitting subgroup and all normal semisimple subgroups, then ''M''/''Z''(''H'') must be trivial, so ''H'' contains its own centralizer. The generalized Fitting subgroup is the smallest subgroup that contains the Fitting subgroup and all normal semisimple subgroups. The generalized Fitting subgroup can also be viewed as a generalized centralizer of chief factors. A nonabelian semisimple group cannot centralize itself, but it does act on itself as inner automorphisms. A group is said to be quasi-nilpotent if every element acts as an inner automorphism on every chief factor. The generalized Fitting subgroup is the unique largest subnormal quasi-nilpotent subgroup, and is equal to the set of all elements which act as inner automorphisms on every chief factor of the whole group : :\operatorname^*(G) = \bigcap\. Here an element ''g'' is in ''H''C''G''(''H''/''K'') if and only if there is some ''h'' in ''H'' such that for every ''x'' in ''H'', ''x''''g'' ≡ ''x''''h'' mod ''K''.


Properties

If ''G'' is a finite solvable group, then the Fitting subgroup contains its own centralizer. The centralizer of the Fitting subgroup is the center of the Fitting subgroup. In this case, the generalized Fitting subgroup is equal to the Fitting subgroup. More generally, if ''G'' is a finite group, then the generalized Fitting subgroup contains its own centralizer. This means that in some sense the generalized Fitting subgroup controls ''G'', because ''G'' modulo the centralizer of ''F''*(''G'') is contained in the automorphism group of ''F''*(''G''), and the centralizer of ''F''*(''G'') is contained in ''F''*(''G''). In particular there are only a finite number of groups with given generalized Fitting subgroup.


Applications

The normalizers of nontrivial ''p''-subgroups of a finite group are called the ''p''-local subgroups and exert a great deal of control over the structure of the group (allowing what is called
local analysis In mathematics, the term local analysis has at least two meanings, both derived from the idea of looking at a problem relative to each prime number ''p'' first, and then later trying to integrate the information gained at each prime into a 'global' ...
). A finite group is said to be of characteristic ''p'' type if ''F''*(''G'') is a ''p''-group for every ''p''-local subgroup, because any
group of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phras ...
defined over a field of characteristic ''p'' has this property. In the
classification of finite simple groups In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else it ...
, this allows one to guess over which field a simple group should be defined. Note that a few groups are of characteristic ''p'' type for more than one ''p''. If a simple group is not of Lie type over a field of given characteristic ''p'', then the ''p''-local subgroups usually have components in the generalized Fitting subgroup, though there are many exceptions for groups that have small rank, are defined over small fields, or are sporadic. This is used to classify the finite simple groups, because if a ''p''-local subgroup has a known component, it is often possible to identify the whole group . The analysis of finite simple groups by means of the structure and embedding of the generalized Fitting subgroups of their maximal subgroups was originated by Helmut Bender and has come to be known as
Bender's method In group theory, Bender's method is a method introduced by for simplifying the local group theoretic analysis of the odd order theorem. Shortly afterwards he used it to simplify the Walter theorem on groups with abelian Sylow 2-subgroups , and Gor ...
. It is especially effective in the exceptional cases where components or
signalizer functor In mathematics, a signalizer functor gives the intersections of a potential subgroup of a finite group with the centralizers of nontrivial elements of an abelian group. The signalizer functor theorem gives conditions under which a signalizer functo ...
s are not applicable.


References

* * * * * {{Citation , last1=Huppert , first1=Bertram , author1-link=Bertram Huppert , last2=Blackburn , first2=Norman , title=Finite groups. III. , publisher=Springer-Verlag , location=Berlin-New York , series=Grundlehren der Mathematischen Wissenschaften , isbn=3-540-10633-2 , mr=0650245 , year=1982 , volume=243 Finite groups Functional subgroups