Fitting's theorem

Last updated April 01, 2019

Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows:

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, which is experimental.

Hans Fitting was a mathematician who worked in group theory. He proved Fitting's theorem and Fitting's lemma, and defined the Fitting subgroup in finite group theory and the Fitting decomposition for Lie algebras and Fitting ideals in ring theory.

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 is nilpotent of class n, then MN is nilpotent of class at most m + n.

By induction it follows also that the subgroup generated by a finite collection of nilpotent normal subgroups is nilpotent. This can be used to show that the Fitting subgroup of certain types of groups (including all finite groups) is nilpotent. However, a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent.

Mathematical induction form of mathematical proof

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 0, 1, 2, 3, and so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder:

Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung and that from each rung we can climb up to the next one.

In mathematics, especially in the area of algebra known as group theory, the Fitting subgroupF 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 subgroupF^*, which is generated by the Fitting subgroup and the components of G.

In abstract algebra, a finite group is a mathematical group with a finite number of elements. A group is a set of elements together with an operation which associates, to each ordered pair of elements, an element of the set. In the case of a finite group, the set is finite.

Order-theoretic statement

In terms of order theory, (part of) Fitting's theorem can be stated as:

Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary.

The set of nilpotent normal subgroups form a lattice of subgroups.

Thus the nilpotent normal subgroups of a finite group also form a bounded lattice, and have a top element, the Fitting subgroup.

However, nilpotent normal subgroups do not in general form a complete lattice, as a subgroup generated by an infinite collection of nilpotent normal subgroups need not be nilpotent, though it will be normal. The join of all nilpotent normal subgroups is still defined as the Fitting subgroup, but it need not be nilpotent.

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Complete lattices appear in many applications in mathematics and computer science. Being a special instance of lattices, they are studied both in order theory and universal algebra.

External links

Fitting's Theorem at PlanetMath.org .

This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

Related Research Articles

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers. They are named after early 19th century mathematician Niels Henrik Abel.

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves the group structure. For example, the cyclic group of addition modulo n can be obtained from the integers by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.

In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order. That is, for each element g of a p-group, there exists a nonnegative integer n such that the product of pⁿ copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.

Generating set of a group Subset of a group such that all group elements can be expressed by finitely many group operations on its elements

In abstract algebra, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their inverses.

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 in the trivial subgroup.

A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with {1}.

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination of finitely many elements of the finite set S and of inverses of such elements.

In mathematics, specifically group theory, a subgroup series is a chain of subgroups:

In mathematics, the lattice of subgroups of a group $is the lattice whose elements are the subgroups of, with the partial order relation being set inclusion. In this lattice, the join of two subgroups is the subgroup generated by their union, and the meet of two subgroups is their intersection.$

In mathematics, a matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication, and a linear group is an abstract group that is isomorphic to a matrix group over a field K, in other words, admitting a faithful, finite-dimensional representation over K.

In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups. Polycyclic groups are finitely presented, and this makes them interesting from a computational point of view.

In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. The concept is credited to work in the 1930s by Russian mathematician Sergei Chernikov.

In mathematics, a group is supersolvable if it has an invariant normal series where all the factors are cyclic groups. Supersolvability is stronger than the notion of solvability.

In mathematics, in the realm of group theory, a countable group is said to be SQ-universal if every countable group can be embedded in one of its quotient groups. SQ-universality can be thought of as a measure of largeness or complexity of a group.

In mathematics, especially in the area of abstract algebra which studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup H≤G such that H has property P.

In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each divisor. This result has been called the fundamental theorem of cyclic groups.

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, this is an explicit expression that the group is a nilpotent group, and for matrix rings, this is an explicit expression that in some basis the matrix ring consists entirely of upper triangular matrices with constant diagonal.

Lattice (discrete subgroup) discrete subgroup in a locally compact topological group

In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of Rⁿ, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood.