Bankruptcy blog

In group theory, a branch of mathematics, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the 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 remainder of this article deals exclusively with finite groups.

The Fitting subgroup

The existence of the Fitting subgroup is guaranteed by Fitting’s theorem which says that the product of a 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 solvable group (not equal to the trivial group), then the Fitting subgroup is always non-trivial, i.e. F(G)≠ 1 (compare with the statement below that the Fitting subgroup contains its own centralizer in G).

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.

Properties

If G is solvable, then the Fitting subgroup contains its own centralizer,
and is the same as the generalized Fitting subgroup.
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

A 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
what field a simple group should be defined over. (But 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 or 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.

Further reading

• M.Aschbacher, Finite group theory, ISBN 0-521-45826-9

• Google SketchUp - Home Developed for the conceptual stages of design, Google SketchUp is a powerful yet easy-to-learn 3D software tool that combines a simple, yet robust tool-set
• iPrioritize - get organized with simple to-do lists Simple to-do lists that can be edited at any time from any place in the world. Email, print, check from your mobile phone, subscribe via RSS, and share with
• Scrap Girls: Digital Scrapbooking is Simple to Learn and Fun to Do Digital scrapbooking made simple. Digital scrapbooking tutorials, freebies, online classes, kits, brushes, templates, layout ideas, free digital
• VIP Simple To Do List free download. VIP Simple To Do List - is a VIP Simple To Do List free download. VIP Simple To Do List - is a fast and convenient tool for household, students, teachers, and everyone who wants to
• Simple Sharing Extensions for Atom and RSS The scope of Simple Sharing Extensions (SSE) is to define the minimum extensions necessary to enable loosely-cooperating applications to use XML-based
• Adbusters : The Magazine - #73 Carbon Neutral Culture / The Simple The Simple Life: How To Bring The Land Back To Us. From Adbusters #73, Sep-Oct 2007. Photo Credit: Kozydan. My mom is confessing to me over the phone.
• Raible Designs | [TSE] Spring-OSGI with Adrian Colyer If you're using Maven, it's simple to switch containers with profiles (i.e. Adrian did a demo showing a simple "Hello World" example deployed into an
• Introduction to the Simple Network Management Protocol (SNMP) Part 1 In this two part article we will look at how to use SNMP, the Simple Network Management Protocol, and install the service on Windows Server 2003.
• Waterstones.com: Food and Drink: Slow Cooking: 150 Delicious Food and Drink: Slow Cooking: 150 Delicious Simple-to-make Recipes Shown in 200 Stunning Photographs - Soups, Stews, Casseroles, Roasts, Comforting Hot-pots
• Harvard, on How to Cope with things Simple to Chaotic « FoundRead It’s about decision making in the context of business situations that range from simple, to complex, complicated or chaotic. “Wise executives tailor their
• Usability In The News It was as simple as using two fingers on the track pad and hitting that same Using two fingers instead of one for a right click was just too simple for
• A VIP Simple To Do List v.2.90 Download - Easy tool for planning VIP Simple To Do List is an easy-to-use PDA software which uses To Do List method to help you do more tasks spending less time. It increases your personal
• elearnspace: Simple to complex thinking Simple to complex thinking Today, during my presentation at McGraw-Hill Ryerson conference at Mount Royal College (Calgary), I was asked about whether we
• Single Transferable Vote STV - It's simple to vote. At elections for District Health Boards and some local It is simple to vote with STV. Instead of putting a tick beside the
• From Simple to Obvious at Like It Matters I think we’ve made a fetish of ’simple’ software. I like the Einstein quote above because it puts simplicity in context. Simplicity is not a value in & of
• Backreaction: A Theoretically Simple Exception of Everything As much as I like the first part, I find this construction neither simple nor particularly beautiful. That is to say, I admittedly don't understand why it
• 1-language.com - Online English Course - Unit 2 Verb "to be Verb "to be" Present Simple. 1. Presentation - Dialogues 2. Grammar Point 3. Grammar Exercises 1 4. Grammar Exercises 2 5. Dialogue Exercises.
• Free Jewish MP3 Torah Audio Downloads | Judaism Free Jewish mp3s to download - interesting topics.