﻿

# 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. 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 Fitting subgroup

The nilpotency of the Fitting subgroup of a finite group is guaranteed by Fitting's theorem 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. Since the Fitting subgroup of a finite solvable group contains its own centralizer, this gives a method of understanding finite solvable groups as extensions of nilpotent groups by faithful automorphism groups of nilpotent groups.

In a nilpotent group, every chief factor 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 harv|Huppert|1967|loc=Kap.VI, Satz 5.4, p.686::The generalization to p-nilpotent groups is similar.

The generalized Fitting subgroup

A component of a group is a subnormal quasisimple 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 groups.

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.

The definition of the generalized Fitting subgroup looks a little strange at first. To motivate it, consider the problem of trying to find 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 "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 groups 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 one 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 harv|Huppert|1967|loc=Kap.VI, Satz 5.4, p. 686::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 any finite group, 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). 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 defined over a field of characteristic "p" has this property. In the classification of finite simple groups, 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 harv|Aschbacher|Seitz|1976.

References

*Citation|last1=Aschbacher|first1=Michael|author1-link=Michael Aschbacher|title=Finite Group Theory|publisher=Cambridge University Press|year=2000|isbn=978-0-521-78675-1
*Citation|last1=Aschbacher|first1=Michael|author1-link=Michael Aschbacher|last2=Seitz|first2=Gary M.|title=On groups with a standard component of known type|journal=Osaka J. Math.|volume=13|year=1976|number=3|pages=439–482
*Citation | last1=Huppert | first1=B. | author1-link=Bertram Huppert | title=Endliche Gruppen | publisher=Springer-Verlag | location=Berlin, New York | language=German | isbn=978-3-540-03825-2 | oclc=527050 | id=MathSciNet | id = 0224703 | year=1967

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• 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, moreover, M is nilpotent of class m and N …   Wikipedia

• 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 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… …   Wikipedia

• Subgroup — This article is about the mathematical concept For the galaxy related concept, see Galaxy subgroup. Concepts in group theory category of groups subgroups, normal subgroups group homomorphisms, kernel, image, quotient direct product, direct sum …   Wikipedia

• Subgroup series — In mathematics, a subgroup series is a chain of subgroups: Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important… …   Wikipedia

• Frattini subgroup — Hasse diagram of the lattice of subgroups of the dihedral group Dih4 In the 3 element layer are the maximal subgroups; their intersection (the F. s.) is the central element in the 5 element layer. So Dih4 has only one non generating element… …   Wikipedia

• Hans Fitting — (13 November 1906 München Gladbach (now Mönchengladbach) – 15 June 1938 Königsberg (now Kaliningrad))was a mathematician who worked in group theory. He proved Fitting s theorem and Fitting s lemma, and defined the Fitting subgroupin finite group… …   Wikipedia

• Descendant subgroup — In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its… …   Wikipedia

• Classification of finite simple groups — Group theory Group theory …   Wikipedia

• List of group theory topics — Contents 1 Structures and operations 2 Basic properties of groups 2.1 Group homomorphisms 3 Basic types of groups …   Wikipedia