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Composition series

In abstract algebra, a composition series is a finite chain of subgroups of a group G, denoted G = G_0 \triangleright G_1 \triangleright \cdots \triangleright G_n = \{e\}, where each G_i is normal in G_{i-1} and the quotient groups G_{i-1}/G_i are simple groups, meaning they have no nontrivial normal subgroups.^[1] Similarly, for a module M over a ring R, a composition series is a chain M = M_0 \supset M_1 \supset \cdots \supset M_n = 0 of submodules where each M_i is invariant under the action of R in M_{i-1}, and the quotients M_{i-1}/M_i are simple modules, possessing no proper nontrivial submodules.^[2] These series provide a way to decompose complex algebraic structures into irreducible building blocks, analogous to prime factorizations in number theory. Every finite group admits a composition series, and by the Jordan-Hölder theorem, any two such series for the same group have isomorphic factor groups up to permutation and repetition.^[3] For modules, the existence of a composition series is equivalent to the module being both Artinian and Noetherian, ensuring finite length, and the Jordan-Hölder theorem extends to guarantee that composition factors are unique up to isomorphism.^[4] Composition series are fundamental in classifying finite groups 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 solvable group.^[5] The concept originated in the study of finite groups but generalizes to broader algebraic settings, including representations of Lie algebras and chain complexes, where it aids in understanding indecomposability and homological properties.^[6]

General Framework

Definition

In abstract algebra, a composition series provides a way to decompose certain algebraic objects into irreducible building blocks. For an object such as a module M over a ring, or more generally, an object with a lattice of subobjects, a composition series is a finite descending chain of subobjects M = M_0 \supset M_1 \supset \cdots \supset M_n = 0 such that each quotient M_{i-1}/M_i is a simple object for i = 1, \dots, n.^[4] This means that no proper nontrivial subobject exists between consecutive terms in the chain, ensuring the series cannot be refined further by inserting additional subobjects.^[1] 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 simple group has no normal subgroups other than the trivial group and itself.^[7] Formally, in the lattice L of subobjects ordered by inclusion, the chain satisfies L_0 = L \supset L_1 \supset \cdots \supset L_n = \{0\} with each factor L_{i-1}/L_i simple, where the quotient structure admits no intermediate subobjects.^[4] Such series are defined analogously in various algebraic settings, including groups and rings, where the subobjects are subgroups or ideals, respectively.^[1] The concept of a composition series motivates the study of algebraic structures by allowing their decomposition into simple components, much like the prime factorization of an integer breaks it down into irreducible primes, revealing the "atomic" structure underlying more complex entities.^[5] This decomposition facilitates understanding invariants and classifications, with the Jordan–Hölder theorem providing a uniqueness result for such series in later developments.^[5]

Basic Properties

A composition series of an algebraic object consists of a finite chain of subobjects L = L_0 \supset L_1 \supset \cdots \supset L_n = 0 such that each quotient L_{i-1}/L_i is simple for i = 1, \dots, n. One fundamental property is the invariance of the length: all composition series of a given object have the same length n, 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.^[8] The composition factors of the series are the simple quotients L_{i-1} / L_i, considered up to isomorphism and permutation. These factors capture the essential structural components of the object, and any two composition series yield the same multiset of composition factors up to isomorphism. This uniqueness up to permutation distinguishes composition series from more general chains, emphasizing their role in decomposition.^[9] The Jordan–Hölder theorem, whose proof relies on the Zassenhaus lemma 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 isomorphism and permutation. In the context of groups, the Zassenhaus lemma states that if A \supseteq A' and B \supseteq B' are subgroups with A' normal in A and B' normal in B, then there is an isomorphism 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.^[8]^[9] 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.^[10]

For Groups

Existence and Examples

Every finite group possesses a composition series due to the finite descending chain condition on normal subgroups.^[11] This condition ensures that starting from the group G, one can iteratively select a maximal proper normal subgroup N_1 \trianglelefteq G, then a maximal proper normal subgroup N_2 \trianglelefteq N_1, and continue this process until reaching the trivial subgroup \{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.^[11] A concrete construction is illustrated by the symmetric group S_3, which has order 6 and admits the composition series S_3 \triangleright A_3 \triangleright \{e\}, where A_3 is the alternating subgroup of order 3.^[11] The successive quotients are S_3 / A_3 \cong C_2 and A_3 / \{e\} \cong C_3, both simple abelian groups of prime order.^[11] Similarly, the alternating group A_4 of order 12 has a composition series A_4 \triangleright V_4 \triangleright \langle (1\,2)(3\,4) \rangle \triangleright \{e\}, where V_4 is the Klein four-subgroup \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\} and \langle (1\,2)(3\,4) \rangle is a subgroup of order 2.^[12] The quotients are A_4 / V_4 \cong C_3, V_4 / \langle (1\,2)(3\,4) \rangle \cong C_2, and \langle (1\,2)(3\,4) \rangle / \{e\} \cong C_2, all simple abelian groups.^[12] 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 if and only if all its composition factors are abelian, hence cyclic of prime order.^[11] For instance, both S_3 and A_4 are solvable, as their factors are abelian, whereas nonsolvable groups like A_5 feature nonabelian simple factors.^[11]

Jordan–Hölder Theorem

The Jordan–Hölder theorem asserts that for a finite group G, any two composition series have the same length, and their composition factors are isomorphic up to permutation and ordering.^[13] This means that the multiset of simple groups appearing as successive quotients in the series is uniquely determined by G, providing a canonical decomposition into simple building blocks.^[13] The theorem's origins trace to Camille Jordan, who introduced composition series around 1870 and proved in 1869 that composition lengths and quotient orders are invariant up to permutation.^[14] Otto Hölder completed the proof in 1889, showing the factors are isomorphic, initially focusing on p-groups using Sylow theorems and classifications for orders like p^3 and p^4.^[15] 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 Zassenhaus lemma (also known as the butterfly lemma).^[13] This theorem plays a central role in representation theory, as the composition factors determine the structure of irreducible representations and aid in classifying group extensions.^[14] 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 \{G_i\} and \{H_j\} of G, there exist refinements \{A_k\} and \{B_\ell\} obtained by inserting intersections G_i \cap H_j and joins G_i H_j. The Zassenhaus lemma then establishes isomorphisms between corresponding subquotients: for normal subgroups A \trianglelefteq G_i and B \trianglelefteq H_j, it yields G_i / (G_i \cap H_j) \cong (G_i H_j)/H_j and dual forms via the second isomorphism theorem, ensuring the refinements have isomorphic factors.^[13] The correspondence theorem further links these to show that the composition factors of the original series match those of the refinements, up to isomorphism and multiplicity, by induction on series length.^[13] 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.^[13] 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.^[1]

For Modules

Definition and Existence

In module theory, a composition series for an R-module M, where R is a ring, is a finite descending chain of submodules

M = M_0 \supset M_1 \supset \cdots \supset M_n = 0

such that each successive quotient M_{i-1}/M_i is a simple R-module for i = 1, \dots, n.^[16] A simple R-module is a nonzero module that admits no proper nonzero submodules.^[16] Examples of simple modules include R/\mathfrak{m} where \mathfrak{m} is a maximal left ideal of R.^[16] The existence of a composition series for a module M is equivalent to M satisfying both the ascending chain condition (Noetherian) and the descending chain condition (Artinian) on submodules, in which case M is said to have finite length.^[16] To construct such a series when it exists, begin with M_0 = M and iteratively select M_{i} as a maximal proper submodule of M_{i-1}; the Artinian condition ensures that this process terminates at zero after finitely many steps.^[17] The length \ell(M) of a module M with a composition series is defined as the number n of nonzero quotients in the series (i.e., the number of simple factors).^[16] This length is well-defined, independent of the choice of series.^[16] As a concrete example, consider finite-dimensional vector spaces over a field k, which are k-modules. Here, the simple k-modules are one-dimensional spaces isomorphic to k itself, and any composition series for a space V of dimension d has length d, corresponding to a flag of subspaces with one-dimensional quotients that reflects the dimension of V.^[18]

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.^[19] 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.^[20] 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 length, the composition factors of a finite-length module are unique up to isomorphism and multiplicity.^[20] This uniqueness extends from the Jordan–Hölder theorem and is complemented by the Krull–Schmidt theorem, which provides a canonical direct sum decomposition into indecomposable summands. The Krull–Schmidt theorem states that if a finite-length module M admits decompositions M \cong U_1 \oplus \cdots \oplus U_m and M \cong V_1 \oplus \cdots \oplus V_n into indecomposable modules, then m = n, and there is a permutation \sigma such that U_i \cong V_{\sigma(i)} for all i.^[21] The composition factors of the module are then the union of the composition factors of these indecomposables. In representation theory, 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 isomorphism. For example, non-isomorphic modules can share identical composition factors, as seen in the case of \mathbb{Z}/4\mathbb{Z} and \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}.^[22]^[23] Modules without finite length, 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.^[20] For example, the rational numbers \mathbb{Q} as a \mathbb{Z}-module has no composition series, since it is neither Artinian nor Noetherian.^[24]

Generalizations

In Abelian Categories

In an abelian category \mathcal{A}, a composition series for an object A is a finite chain of subobjects

$0 = A_0 \subset A_1 \subset \cdots \subset A_n = A

such that each successive quotient A_i / A_{i-1} is a simple object, i.e., nonzero with no proper nonzero subobjects.^[25] An object A \in \mathcal{A} admits a composition series if and only if it has finite length, meaning both Artinian (descending chains of subobjects stabilize) and Noetherian (ascending chains stabilize).^[26] In categories of finite length objects, such as the category of finite-length modules over an Artinian ring (a special case), every object has a composition series constructed by iteratively extracting maximal proper subobjects.^[26] The Jordan–Hölder theorem holds in any abelian category: for a finite-length object, all composition series have the same length n, and there is a permutation \sigma such that the simple quotients satisfy A_i / A_{i-1} \cong B_{\sigma(i)} / B_{\sigma(i)-1} for any two series $0 = B_0 \subset \cdots \subset B_n = A. The proof relies on refining arbitrary subobject 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.^[27] 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.^[28] 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.^[29] 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.^[30]

Other Algebraic Structures

In ring theory, a composition series for a ring R consists of a finite chain of left ideals R = I_0 \supsetneq I_1 \supsetneq \cdots \supsetneq I_n = 0 such that each successive quotient I_k / I_{k+1} is a simple left module over R / I_{k+1}.^[31] Such series exist precisely when R is Artinian, meaning it satisfies the descending chain condition on ideals, and the length of the series equals the composition length of R as a module over itself.^[32] 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 \cong M_{n_1}(D_1) \times \cdots \times M_{n_r}(D_r), where each D_i is a division ring and n_i \geq 1.^[32] This structure arises because the minimal left ideals form a composition series as a direct sum of simple modules.^[32] In lattice theory, a composition series is a maximal chain in a lattice L from the bottom element $0 to the top element $1, where each consecutive pair consists of covering elements, and the successive quotients are simple (atoms or coatoms).^[33] Such series exist in lattices of finite length, particularly modular lattices, which satisfy the modular law: for x \leq z, x \vee (y \wedge z) = (x \vee y) \wedge z.^[33] 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.^[33] Distributive lattices, a subclass where the distributive law holds (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.^[33] The height function h(x) in these lattices measures the length of a maximal chain from $0 to x, providing the rank or dimension, and satisfies h(x \vee y) + h(x \wedge y) \leq h(x) + h(y) in semi-modular cases.^[33] For example, the subgroup lattice of a finite group is modular, and its composition series correspond to chief series with simple abelian factors.^[34] For Lie algebras, the analog of a composition series is a chief series, a finite chain of ideals \mathfrak{g} = \mathfrak{g}_0 \triangleright \mathfrak{g}_1 \triangleright \cdots \triangleright \mathfrak{g}_n = 0 where each factor \mathfrak{g}_i / \mathfrak{g}_{i+1} is irreducible as a \mathfrak{g}-module (chief factor), meaning it has no nontrivial submodules or quotients beyond itself and zero.^[35] 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.^[36] The Jordan–Hölder theorem applies, ensuring that chief series have the same length and isomorphic factors up to permutation and extension by scalars.^[35] For instance, in the Heisenberg algebra over \mathbb{C}, a chief series has length 3 with abelian factors of dimensions 1, 1, and 1.^[37] 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.^[34] In any poset, the Hausdorff maximality principle guarantees the existence of maximal chains extending any given chain, assuming the axiom of choice.^[34] 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.^[38] In finite posets, Dilworth's theorem relates the size of the largest antichain to the minimal number of chains covering the poset, where maximal chains help decompose the structure.^[34] For example, in the poset of subspaces of a vector space, maximal chains are flags with simple quotients being lines.^[38] Recent developments since 2000 have extended composition series to triangulated and derived categories, particularly in homotopy theory and algebraic geometry. In the derived category D^b(\mathcal{A}) of a finite-length abelian category \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.^[39] For quasi-hereditary algebras, derived categories admit composition series of varying lengths depending on the tilting structure, with factors being derived categories of simples.^[40] In stable homotopy theory, post-2000 work on smash products and completions in E-based Adams spectral sequences uses derived completions to build filtration towers analogous to chief series, resolving homotopy groups via composition factors.^[41] 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.^[42]

References

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A composition series for G is a normal series such that each factor is simple, i.e., each factor is either cyclic of prime order or a simple nonabelian group.
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Example S4 has the following composition series of length 4, where K is the Klein group {(1), (12)(34), (13)(24), (14)(23)}. S4 > A4 > K > h(12)(34)i > {1}. We ...
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