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Composition series
Composition series
Composition series
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In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group 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 of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules 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 classes 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 groups 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.

For groups

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If a group G has a normal subgroup 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. Otherwise, the question naturally arises as to whether G can be reduced to simple "pieces", and if so, whether there are any unique features of the way this can be done.

More formally, a composition series of a group G is a subnormal series of finite length

with strict inclusions, such that each Hi is a maximal proper normal subgroup of Hi+1. Equivalently, a composition series is a subnormal series such that each factor group Hi+1 / Hi is simple. The factor groups are called composition factors.

A subnormal series is a composition series if and only if 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 has a composition series, but not every infinite group has one. For example, has no composition series.

Uniqueness: Jordan–Hölder theorem

[edit]

A group may have more than one composition series. However, the Jordan–Hölder theorem (named after Camille Jordan and Otto Hölder) 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 permutation and isomorphism. This theorem can be proved using the Schreier refinement theorem. The Jordan–Hölder theorem is also true for transfinite ascending composition series, but not transfinite descending composition series (Birkhoff 1934). Baumslag (2006) gives a short proof of the Jordan–Hölder theorem by intersecting the terms in one subnormal series with those in the other series.

Example

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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.

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

[edit]
See also: Length of a module

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 Jk is a maximal submodule of Jk+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 Jk+1/Jk 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[1] that a module has a finite composition series if and only if it is both an Artinian module and a Noetherian module. If R is an Artinian ring, 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

[edit]

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 (Bourbaki 1974, Ch. 1) or (Isaacs 1994, Ch. 10), 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 automorphisms. 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

[edit]

A composition series of an object A in an abelian category is a sequence of subobjects

such that each quotient object Xi /Xi + 1 is simple (for 0 ≤ i < n). If A has a composition series, the integer n only depends on A and is called the length of A.[2]

See also

[edit]

Notes

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References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Composition series

View on Grokipedia
from Grokipedia
In , a composition series is a finite chain of subgroups of a group GG, denoted G=G0▹G1▹⋯▹Gn={e}G = G_0 \triangleright G_1 \triangleright \cdots \triangleright G_n = \{e\}, where each GiG_i is normal in Gi−1G_{i-1} and the groups Gi−1/GiG_{i-1}/G_i are simple groups, meaning they have no nontrivial normal subgroups. Similarly, for a module MM over a ring RR, a composition series is a chain M=M0⊃M1⊃⋯⊃Mn=0M = M_0 \supset M_1 \supset \cdots \supset M_n = 0 of submodules where each MiM_i is invariant under of RR in Mi−1M_{i-1}, and the quotients Mi−1/MiM_{i-1}/M_i are simple modules, possessing no proper nontrivial submodules. These series provide a way to decompose complex algebraic structures into irreducible building blocks, analogous to prime factorizations in . Every admits a composition series, and by the Jordan-Hölder , any two such series for the same group have isomorphic factor groups up to and repetition. For modules, the existence of a composition series is equivalent to the module being both Artinian and Noetherian, ensuring finite , and the Jordan-Hölder extends to guarantee that composition factors are unique up to . Composition series are fundamental in classifying s and modules, revealing their structure through simple constituents, and play a key role in solvability criteria, such as when all factors are abelian (cyclic of prime order), indicating a . The concept originated in the study of finite groups but generalizes to broader algebraic settings, including representations of algebras and complexes, where it aids in understanding indecomposability and homological properties.

General Framework

Definition

In , a composition series provides a way to decompose certain algebraic objects into irreducible building blocks. For an object such as a module MM over a ring, or more generally, an object with a lattice of subobjects, a composition series is a finite descending of subobjects M=M0⊃M1⊃⋯⊃Mn=0M = M_0 \supset M_1 \supset \cdots \supset M_n = 0 such that each quotient Mi−1/MiM_{i-1}/M_i is a simple object for i=1,…,ni = 1, \dots, n. This means that no proper nontrivial subobject exists between consecutive terms in the , ensuring the series cannot be refined further by inserting additional subobjects. A simple object, in this context, is one that possesses no nontrivial proper subobjects; for example, a simple module has only the zero submodule and itself as submodules, while a has no normal subgroups other than the and itself. Formally, in the lattice LL of subobjects ordered by inclusion, the chain satisfies L0=L⊃L1⊃⋯⊃Ln={0}L_0 = L \supset L_1 \supset \cdots \supset L_n = \{0\} with each factor Li−1/LiL_{i-1}/L_i simple, where the admits no intermediate subobjects. Such series are defined analogously in various algebraic settings, including groups and rings, where the subobjects are subgroups or ideals, respectively. The concept of a composition series motivates the study of algebraic structures by allowing their into simple components, much like the prime of an breaks it down into irreducible primes, revealing the "atomic" structure underlying more complex entities. This facilitates understanding invariants and classifications, with the Jordan–Hölder theorem providing a uniqueness result for such series in later developments.

Basic Properties

A composition series of an algebraic object consists of a finite of subobjects L=L0⊃L1⊃⋯⊃Ln=0L = L_0 \supset L_1 \supset \cdots \supset L_n = 0 such that each Li−1/LiL_{i-1}/L_i is simple for i=1,…,ni = 1, \dots, n. One fundamental property is the invariance of the length: all composition series of a given object have the same length nn, referred to as the composition length of the object. This invariance ensures that the series provides a consistent measure of the object's complexity in terms of simple building blocks. The composition factors of the series are the simple quotients Li−1/LiL_{i-1} / L_i, considered up to and . These factors capture the essential structural components of the object, and any two composition series yield the same of composition factors up to . This uniqueness up to distinguishes composition series from more general chains, emphasizing their role in decomposition. The Jordan–Hölder theorem, whose proof relies on the and the Schreier refinement theorem for subnormal series, establishes that any two composition series have the same length and the same composition factors up to and . In the context of groups, the states that if A⊇A′A \supseteq A' and B⊇B′B \supseteq B' are subgroups with A′A' normal in AA and B′B' normal in BB, then there is an A(A′∩B)/A(A′∩B′)≅B(B′∩A)/B(B′∩A′)A(A' \cap B)/A(A' \cap B') \cong B(B' \cap A)/B(B' \cap A'); analogous versions hold in module lattices and other settings with suitable normality conditions. The lemma ensures that intersections and generated subobjects align isomorphically, enabling the proof that series are equivalent. The notion of composition series originated in the late 19th century through the work of Camille Jordan and Otto Hölder on finite groups, where Jordan established the invariance of factor orders around 1870 and Hölder proved the full isomorphism result in 1889; the concept was subsequently generalized to modules and other algebraic structures.

For Groups

Existence and Examples

Every finite group possesses a composition series due to the finite descending chain condition on normal subgroups. This condition ensures that starting from the group GG, one can iteratively select a maximal proper normal subgroup N1⊴GN_1 \trianglelefteq G, then a maximal proper normal subgroup N2⊴N1N_2 \trianglelefteq N_1, and continue this process until reaching the trivial subgroup {e}\{e\}, yielding a finite chain where each quotient is simple. The Jordan–Hölder theorem then guarantees that any two such series have the same length and isomorphic composition factors up to permutation. A concrete construction is illustrated by the symmetric group S3S_3, which has order 6 and admits the composition series S3▹A3▹{e}S_3 \triangleright A_3 \triangleright \{e\}, where A3A_3 is the alternating subgroup of order 3. The successive quotients are S3/A3≅C2S_3 / A_3 \cong C_2 and A3/{e}≅C3A_3 / \{e\} \cong C_3, both simple abelian groups of prime order. Similarly, the alternating group A4A_4 of order 12 has a composition series A4▹V4▹⟨(1 2)(3 4)⟩▹{e}A_4 \triangleright V_4 \triangleright \langle (1\,2)(3\,4) \rangle \triangleright \{e\}, where V4V_4 is the Klein four-subgroup {e,(1 2)(3 4),(1 3)(2 4),(1 4)(2 3)}\{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\} and ⟨(1 2)(3 4)⟩\langle (1\,2)(3\,4) \rangle is a subgroup of order 2. The quotients are A4/V4≅C3A_4 / V_4 \cong C_3, V4/⟨(1 2)(3 4)⟩≅C2V_4 / \langle (1\,2)(3\,4) \rangle \cong C_2, and ⟨(1 2)(3 4)⟩/{e}≅C2\langle (1\,2)(3\,4) \rangle / \{e\} \cong C_2, all simple abelian groups. The existence of a composition series holds for all finite groups, but the structure of the simple composition factors distinguishes solvable from nonsolvable groups: a finite group is solvable all its composition factors are abelian, hence cyclic of prime order. For instance, both S3S_3 and A4A_4 are solvable, as their factors are abelian, whereas nonsolvable groups like A5A_5 feature nonabelian simple factors.

Jordan–Hölder Theorem

The Jordan–Hölder theorem asserts that for a GG, any two composition series have the same length, and their composition factors are isomorphic up to and ordering. This means that the of simple groups appearing as successive quotients in the series is uniquely determined by GG, providing a into simple building blocks. The theorem's origins trace to Camille Jordan, who introduced composition series around 1870 and proved in 1869 that composition lengths and orders are invariant up to . Otto Hölder completed the proof in 1889, showing the factors are isomorphic, initially focusing on p-groups using and classifications for orders like p3p^3 and p4p^4. The full modern statement emerged through refinements, including Otto Schreier's 1928 proof via the Schreier refinement theorem and Hans Zassenhaus's 1934 improvement using the (also known as the butterfly lemma). This theorem plays a central role in , as the composition factors determine the structure of irreducible representations and aid in classifying group extensions. The proof proceeds by showing that any two subnormal series can be refined to equivalent composition series. First, the Schreier refinement theorem guarantees that given two subnormal series {Gi}\{G_i\} and {Hj}\{H_j\} of GG, there exist refinements {Ak}\{A_k\} and {Bℓ}\{B_\ell\} obtained by inserting intersections Gi∩HjG_i \cap H_j and joins GiHjG_i H_j. The then establishes between corresponding subquotients: for normal subgroups A⊴GiA \trianglelefteq G_i and B⊴HjB \trianglelefteq H_j, it yields Gi/(Gi∩Hj)≅(GiHj)/HjG_i / (G_i \cap H_j) \cong (G_i H_j)/H_j and dual forms via the second , ensuring the refinements have isomorphic factors. The correspondence further links these to show that the composition factors of the original series match those of the refinements, up to and multiplicity, by induction on series length. A key implication is that finite groups are classified by the multiset of their simple composition factors; no two distinct non-isomorphic simple groups can appear without corresponding multiplicities in another series, mirroring the fundamental theorem of arithmetic for integers. For infinite groups, the theorem extends to those of finite composition length—meaning they possess at least one composition series—where all such series remain equivalent under the same conditions.

For Modules

Definition and Existence

In module theory, a composition series for an RR-module MM, where RR is a ring, is a finite descending chain of submodules M=M0⊃M1⊃⋯⊃Mn=0M = M_0 \supset M_1 \supset \cdots \supset M_n = 0 such that each successive Mi−1/MiM_{i-1}/M_i is a simple RR-module for i=1,…,ni = 1, \dots, n. A simple RR-module is a nonzero module that admits no proper nonzero submodules. Examples of simple modules include R/mR/\mathfrak{m} where m\mathfrak{m} is a maximal left ideal of RR. The existence of a composition series for a module MM is equivalent to MM satisfying both the ascending chain condition (Noetherian) and the descending chain condition (Artinian) on submodules, in which case MM is said to have finite length. To construct such a series when it exists, begin with M0=MM_0 = M and iteratively select MiM_{i} as a maximal proper submodule of Mi−1M_{i-1}; the Artinian condition ensures that this process terminates at zero after finitely many steps. The length ℓ(M)\ell(M) of a module MM with a composition series is defined as the number nn of nonzero quotients in the series (i.e., the number of simple factors). This length is well-defined, independent of the choice of series. As a concrete example, consider finite-dimensional vector spaces over a field kk, which are kk-modules. Here, the simple kk-modules are one-dimensional spaces isomorphic to kk itself, and any composition series for a space VV of dimension dd has length dd, corresponding to a flag of subspaces with one-dimensional quotients that reflects the dimension of VV.

Uniqueness Results

For a module of finite length over an arbitrary ring, all composition series have the same length, defined as the number of simple factors in the series. This follows from the Jordan–Hölder theorem for modules, which asserts that any two composition series of such a module are equivalent: they have the same length, and their successive quotients (the composition factors) are isomorphic up to permutation and isomorphism. Thus, the composition factors determine the simple modules appearing in the series and their multiplicities. Over Artinian rings, where every finitely generated module has finite , the composition factors of a finite- module are unique up to and multiplicity. This uniqueness extends from the Jordan–Hölder theorem and is complemented by the Krull–Schmidt theorem, which provides a decomposition into indecomposable summands. The Krull–Schmidt theorem states that if a finite- module MM admits decompositions M≅U1⊕⋯⊕UmM \cong U_1 \oplus \cdots \oplus U_m and M≅V1⊕⋯⊕VnM \cong V_1 \oplus \cdots \oplus V_n into indecomposable modules, then m=nm = n, and there is a σ\sigma such that Ui≅Vσ(i)U_i \cong V_{\sigma(i)} for all ii. The composition factors of the module are then the union of the composition factors of these indecomposables. In , finite-length modules with the same composition factors (including multiplicities) have the same Brauer character, which provides an invariant but does not determine the module up to . For example, non-isomorphic modules can share identical composition factors, as seen in the case of Z/4Z\mathbb{Z}/4\mathbb{Z} and Z/2Z⊕Z/2Z\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}. Modules without finite , such as infinite-dimensional vector spaces over a field, lack composition series entirely, as they admit strictly ascending chains of subspaces of arbitrary finite length. For example, the rational numbers Q\mathbb{Q} as a Z\mathbb{Z}-module has no composition series, since it is neither Artinian nor Noetherian.

Generalizations

In Abelian Categories

In an abelian category A\mathcal{A}, a composition series for an object AA is a finite chain of subobjects 0=A0⊂A1⊂⋯⊂An=A0 = A_0 \subset A_1 \subset \cdots \subset A_n = A such that each successive quotient Ai/Ai−1A_i / A_{i-1} is a simple object, i.e., nonzero with no proper nonzero subobjects. An object A∈AA \in \mathcal{A} admits a if and only if it has finite , meaning both Artinian (descending chains of subobjects stabilize) and Noetherian (ascending chains stabilize). In categories of finite length objects, such as the category of finite-length modules over an (a special case), every object has a composition series constructed by iteratively extracting maximal proper subobjects. The Jordan–Hölder theorem holds in any : for a finite-length object, all composition series have the same length nn, and there is a σ\sigma such that the simple quotients satisfy Ai/Ai−1≅Bσ(i)/Bσ(i)−1A_i / A_{i-1} \cong B_{\sigma(i)} / B_{\sigma(i)-1} for any two series 0=B0⊂⋯⊂Bn=A0 = B_0 \subset \cdots \subset B_n = A. The proof relies on refining arbitrary chains to composition series via the Artinian/Noetherian properties and comparing refinements using exact sequences and the zigzag lemma, without needing enough projectives or injectives. Composition series find key applications in the category of coherent sheaves on a Noetherian scheme, an abelian category where finite-length coherent sheaves (e.g., those with support of dimension zero) decompose into simple quotients, which are skyscraper sheaves at points, enabling computations of sheaf cohomology and Hilbert polynomials. Similarly, in the abelian category of finite-dimensional modules over a finite-dimensional algebra over an algebraically closed field, every object has a composition series whose simple factors are the irreducible representations, with the Jordan–Hölder theorem ensuring unique multiplicities up to isomorphism, foundational for Auslander–Reiten quiver classifications. A categorical analog of the Artin–Rees lemma appears in the control of filtrations on subobjects: in abelian categories with Noetherian objects, like coherent sheaf categories, the intersection of powers of a filtration (e.g., by a coherent ideal sheaf) stabilizes relative to a fixed subobject, preserving length and support properties in derived categories.

Other Algebraic Structures

In ring theory, a composition series for a ring RR consists of a finite chain of left ideals R=I0⊋I1⊋⋯⊋In=0R = I_0 \supsetneq I_1 \supsetneq \cdots \supsetneq I_n = 0 such that each successive quotient Ik/Ik+1I_k / I_{k+1} is a simple left module over R/Ik+1R / I_{k+1}. Such series exist precisely when RR is Artinian, meaning it satisfies the descending chain condition on ideals, and the length of the series equals the composition length of RR as a module over itself. For semisimple Artinian rings, which admit a composition series with simple factors, Wedderburn's little theorem implies that simple Artinian rings are matrix rings over division rings, while the full Wedderburn–Artin theorem states that any semisimple Artinian ring decomposes uniquely (up to isomorphism and ordering) as a finite direct product R≅Mn1(D1)×⋯×Mnr(Dr)R \cong M_{n_1}(D_1) \times \cdots \times M_{n_r}(D_r), where each DiD_i is a division ring and ni≥1n_i \geq 1. This structure arises because the minimal left ideals form a composition series as a direct sum of simple modules. In lattice theory, a composition series is a maximal chain in a lattice LL from the bottom element 00 to the top element 11, where each consecutive pair consists of covering elements, and the successive quotients are simple (atoms or coatoms). Such series exist in lattices of finite length, particularly modular lattices, which satisfy the modular law: for x≤zx \leq z, x∨(y∧z)=(x∨y)∧zx \vee (y \wedge z) = (x \vee y) \wedge z. In modular lattices, the Jordan–Hölder theorem guarantees that any two composition series have the same length, and their factors are isomorphic up to permutation. Distributive lattices, a subclass where the distributive law holds (a∧(b∨c)=(a∧b)∨(a∧c)a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)), also admit composition series, often represented via Birkhoff's theorem as lattices of down-sets in a poset of join-irreducibles. The height function h(x)h(x) in these lattices measures the length of a maximal chain from 00 to xx, providing the rank or dimension, and satisfies h(x∨y)+h(x∧y)≤h(x)+h(y)h(x \vee y) + h(x \wedge y) \leq h(x) + h(y) in semi-modular cases. For example, the subgroup lattice of a finite group is modular, and its composition series correspond to chief series with simple abelian factors. For Lie algebras, the analog of a composition series is a chief series, a finite chain of ideals g=g0▹g1▹⋯▹gn=0\mathfrak{g} = \mathfrak{g}_0 \triangleright \mathfrak{g}_1 \triangleright \cdots \triangleright \mathfrak{g}_n = 0 where each factor gi/gi+1\mathfrak{g}_i / \mathfrak{g}_{i+1} is irreducible as a g\mathfrak{g}-module (chief factor), meaning it has no nontrivial submodules or quotients beyond itself and zero. Finite-dimensional Lie algebras over an algebraically closed field of characteristic zero always possess chief series, as their solvable radical admits a composition series of derived subalgebras, and semisimple ones decompose into direct sums of simple ideals. The Jordan–Hölder theorem applies, ensuring that chief series have the same length and isomorphic factors up to permutation and extension by scalars. For instance, in the Heisenberg algebra over C\mathbb{C}, a chief series has length 3 with abelian factors of dimensions 1, 1, and 1. Generalizations to partially ordered sets (posets) treat composition series as maximal chains, which are totally ordered subsets that cannot be properly extended while remaining chains. In any poset, the Hausdorff maximality principle guarantees the existence of maximal chains extending any given chain, assuming the . For graded posets, such as the submodule lattice of a module, maximal chains correspond exactly to composition series, with uniform length equal to the rank. In finite posets, relates the size of the largest to the minimal number of chains covering the poset, where maximal chains help decompose the structure. For example, in the poset of subspaces of a , maximal chains are flags with simple quotients being lines. Recent developments since 2000 have extended composition series to triangulated and derived categories, particularly in homotopy theory and algebraic geometry. In the derived category Db(A)D^b(\mathcal{A}) of a finite-length abelian category A\mathcal{A}, a composition series is a chain of admissible subcategories with simple quotients (thick ideals), and the length measures the "dimension" of the category. For quasi-hereditary algebras, derived categories admit composition series of varying lengths depending on the tilting structure, with factors being derived categories of simples. In stable homotopy theory, post-2000 work on smash products and completions in EE-based Adams spectral sequences uses derived completions to build filtration towers analogous to chief series, resolving homotopy groups via composition factors. These categorical enhancements address limitations in classical settings by incorporating homotopy equivalences, as in Orlov's equivalences between derived categories of coherent sheaves on varieties.

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