In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ' ...
, a composition series provides a way to break up an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
, such as 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 ide ...
or a
module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not
semisimple, hence cannot be decomposed into a
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every ...
s. A composition series of a module ''M'' is a finite increasing
filtration
Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
of ''M'' by
submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
s such that the successive quotients are
simple and serves as a replacement of the direct sum decomposition of ''M'' into its simple constituents.
A composition series may not exist, and when it does, it need not be unique. Nevertheless, a group of results known under the general name Jordan–Hölder theorem asserts that whenever composition series exist, the ''
isomorphism class
In mathematics, an isomorphism class is a collection of mathematical objects isomorphic to each other.
Isomorphism classes are often defined as the exact identity of the elements of the set is considered irrelevant, and the properties of the str ...
es'' of simple pieces (although, perhaps, not their ''location'' in the composition series in question) and their multiplicities are uniquely determined. Composition series may thus be used to define invariants of
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 and
Artinian modules.
A related but distinct concept is a
chief series: a composition series is a maximal
''subnormal'' series, while a chief series is a maximal ''
normal series In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups:
:1 = A_0 \leq A_1 \leq \cdots \leq A_n = G
where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simpler ...
''.
For groups
If a group ''G'' has a
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 ...
''N'', then the factor group ''G''/''N'' may be formed, and some aspects of the study of the structure of ''G'' may be broken down by studying the "smaller" groups ''G/N'' and ''N''. If ''G'' has no normal subgroup that is different from ''G'' and from the trivial group, then ''G'' is a
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 ...
. Otherwise, the question naturally arises as to whether ''G'' can be reduced to simple "pieces", and if so, are there any unique features of the way this can be done?
More formally, a composition series 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 ide ...
''G'' is a
subnormal series of finite length
:
with strict inclusions, such that each ''H''
''i'' is a
maximal proper normal subgroup of ''H''
''i''+1. Equivalently, a composition series is a subnormal series such that each factor group ''H''
''i''+1 / ''H''
''i'' is
simple. The factor groups are called composition factors.
A subnormal series is a composition series
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicond ...
it is of maximal length. That is, there are no additional subgroups which can be "inserted" into a composition series. The length ''n'' of the series is called the composition length.
If a composition series exists for a group ''G'', then any subnormal series of ''G'' can be ''refined'' to a composition series, informally, by inserting subgroups into the series up to maximality. Every
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 ...
has a composition series, but not every
infinite group
In group theory, an area of mathematics, an infinite group is a group whose underlying set contains an infinite number of elements. In other words, it is a group of infinite order.
Examples
* (Z, +), the group of integers with addition is in ...
has one. For example,
has no composition series.
Uniqueness: Jordan–Hölder theorem
A group may have more than one composition series. However, the Jordan–Hölder theorem (named after
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
and
Otto Hölder
Ludwig Otto Hölder (December 22, 1859 – August 29, 1937) was a German mathematician born in Stuttgart.
Early life and education
Hölder was the youngest of three sons of professor Otto Hölder (1811–1890), and a grandson of professor Christ ...
) states that any two composition series of a given group are equivalent. That is, they have the same composition length and the same composition factors,
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
and
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. This theorem can be proved using the
Schreier refinement theorem In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups tha ...
. The Jordan–Hölder theorem is also true for
transfinite Transfinite may refer to:
* Transfinite number, a number larger than all finite numbers, yet not absolutely infinite
* Transfinite induction, an extension of mathematical induction to well-ordered sets
** Transfinite recursion
Transfinite inducti ...
''ascending'' composition series, but not transfinite ''descending'' composition series .
gives a short proof of the Jordan–Hölder theorem by intersecting the terms in one subnormal series with those in the other series.
Example
For a
cyclic group of order ''n'', composition series correspond to ordered prime factorizations of ''n'', and in fact yields a proof of the
fundamental theorem of arithmetic
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
.
For example, the cyclic group
has
and
as three different composition series. The sequences of composition factors obtained in the respective cases are
and
For modules
The definition of composition series for modules restricts all attention to submodules, ignoring all additive subgroups that are ''not'' submodules. Given a ring ''R'' and an ''R''-module ''M'', a composition series for ''M'' is a series of submodules
:
where all inclusions are strict and ''J''
''k'' is a maximal submodule of ''J''
''k''+1 for each ''k''. As for groups, if ''M'' has a composition series at all, then any finite strictly increasing series of submodules of ''M'' may be refined to a composition series, and any two composition series for ''M'' are equivalent. In that case, the (simple) quotient modules ''J''
''k''+1/''J''
''k'' are known as the composition factors of ''M,'' and the Jordan–Hölder theorem holds, ensuring that the number of occurrences of each isomorphism type of simple ''R''-module as a composition factor does not depend on the choice of composition series.
It is well known that a module has a finite composition series if and only if it is both an
Artinian module and a
Noetherian module In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the prop ...
. If ''R'' is an
Artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are ...
, then every finitely generated ''R''-module is Artinian and Noetherian, and thus has a finite composition series. In particular, for any field ''K'', any finite-dimensional module for a finite-dimensional algebra over ''K'' has a composition series, unique up to equivalence.
Generalization
Groups with a set of operators generalize group actions and ring actions on a group. A unified approach to both groups and modules can be followed as in or , simplifying some of the exposition. The group ''G'' is viewed as being acted upon by elements (operators) from a set Ω. Attention is restricted entirely to subgroups invariant under the action of elements from Ω, called Ω-subgroups. Thus Ω-composition series must use only Ω-subgroups, and Ω-composition factors need only be Ω-simple. The standard results above, such as the Jordan–Hölder theorem, are established with nearly identical proofs.
The special cases recovered include when Ω = ''G'' so that ''G'' is acting on itself. An important example of this is when elements of ''G'' act by conjugation, so that the set of operators consists of the
inner automorphism
In abstract algebra an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via simple operations from within the group it ...
s. A composition series under this action is exactly a
chief series. Module structures are a case of Ω-actions where Ω is a ring and some additional axioms are satisfied.
For objects in an abelian category
A composition series of an
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
''A'' in an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of a ...
is a sequence of subobjects
:
such that each
quotient object In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theor ...
''X
i'' /''X''
''i'' + 1 is
simple (for ). If ''A'' has a composition series, the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
''n'' only depends on ''A'' and is called the
length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
of ''A''.
See also
*
Krohn–Rhodes theory In mathematics and computer science, the Krohn–Rhodes theory (or algebraic automata theory) is an approach to the study of finite semigroups and automata that seeks to decompose them in terms of elementary components. These components correspond t ...
, a semigroup analogue
*
Schreier refinement theorem In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups tha ...
, any two equivalent
subnormal series have equivalent composition series refinements
*
Zassenhaus lemma Zassenhaus is a German surname. Notable people with the surname include:
* Hans Zassenhaus (1912–1991), German mathematician
** Zassenhaus algorithm
** Zassenhaus group In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certai ...
, used to prove the Schreier Refinement Theorem
Notes
References
*
*
*
*
*{{Citation
, last=Kashiwara
, first=Masaki
, last2=Schapira
, first2=Pierre
, title=Categories and sheaves
, year=2006
Subgroup series
Module theory