Fitting's theorem is a

mathematical 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 ...

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...

proved by Hans Fitting. It can be stated as follows: :If ''M'' and ''N'' are

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 ...

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

s of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

''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

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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 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 smalles ...

of certain types of groups (including all

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) is nilpotent. However, a subgroup generated by an ''infinite'' collection of nilpotent normal subgroups need not be nilpotent.

Order-theoretic statement

In terms of

order theory Order theory is a branch of mathematics that 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 intr ...

, (part of) Fitting's theorem can be stated as: :The set of nilpotent normal subgroups form a

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

. 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 In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally 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.

External links

* Theorems in group theory {{Abstract-algebra-stub