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Cayley's theorem

Cayley's theorem is a fundamental result in abstract algebra stating that every group G is isomorphic to a subgroup of the symmetric group \operatorname{Sym}(G) acting on the underlying set of G. Named after the British mathematician Arthur Cayley (1821–1895), the theorem was first articulated in his seminal 1854 paper "On the theory of groups, as depending on the symbolic equation \theta^n = 1," published in the Philosophical Magazine, which is widely regarded as the inaugural work introducing the abstract concept of a finite group. In this paper, Cayley introduced the representation of groups via permutations, showing a one-to-one correspondence by considering the action of the group on itself via left multiplication, establishing that groups could be studied through their permutation representations without reliance on specific realizations like linear transformations or permutations of geometric objects.^[1] Although Cayley's initial formulation focused on finite groups satisfying certain symbolic equations, the result generalizes to all groups, including infinite ones, via the regular representation.^[2] The proof of Cayley's theorem relies on constructing a homomorphism from G to \operatorname{Sym}(G) defined by \phi_g(h) = gh for all g, h \in G, which is the left regular action; this map is injective because if \phi_g is the identity permutation, then g must be the identity element, ensuring the isomorphism onto its image.^[3] This embedding highlights the universality of permutation groups, as the symmetric group \operatorname{Sym}(G) captures all possible bijections on the set G. For finite groups of order n, it implies an isomorphism to a subgroup of the symmetric group S_n. The theorem's significance lies in its foundational role in reducing the study of arbitrary groups to permutation groups, facilitating computational and structural analysis in group theory; it underpins applications in combinatorics, representation theory, and computer science, such as algorithm design for group computations.^[4] By embedding groups into symmetric groups, it enables the use of permutation-based tools to explore abstract algebraic structures, influencing developments from Burnside's work on enumeration to modern computational group theory software.^[5]

Historical Context

Origins and Discovery

In the mid-19th century, the foundations of group theory were laid amid efforts to understand the solvability of polynomial equations, building on the pioneering work of mathematicians such as Joseph-Louis Lagrange, Paolo Ruffini, Niels Henrik Abel, and Évariste Galois. Galois, in particular, had demonstrated in the 1830s that the solvability of general polynomial equations by radicals is tied to the structure of permutation groups acting on the roots, introducing key concepts like normal subgroups and factor groups that decomposed groups into cosets. This permutation-based approach marked a shift toward abstract algebraic structures, influencing subsequent developments in early abstract algebra.^[6] Arthur Cayley advanced this framework significantly in his 1854 paper, "On the theory of groups, as depending on the symbolic equation θ^n = 1," published in the Philosophical Magazine. Motivated by the study of roots of unity and transformations arising in the solution of polynomial equations, Cayley provided the first explicit abstract definition of a finite group as a set of distinct symbols closed under a binary operation, with an identity element and the operation being associative. He emphasized that such groups arise naturally in algebraic contexts, such as the symmetries of equation solutions, extending beyond the concrete permutations emphasized by Galois.^[5]^[6] In the same paper, Cayley stated that every finite group of order n can be represented faithfully as a subgroup of the symmetric group on n letters via permutations induced by left multiplication, establishing a one-to-one correspondence, though without explicitly verifying the homomorphism property. However, Cayley did not explicitly demonstrate that this correspondence preserves the group operation (i.e., is a homomorphism), a step completed in his later work. This insight unified abstract groups with permutation groups, laying the groundwork for Cayley's theorem while attributing the permutation idea to Galois.^[5]^[7]

Subsequent Developments

Following Arthur Cayley's initial statement of the theorem in 1854, he provided a rigorous proof for finite groups in his 1878 paper "Desiderata and suggestions. No. 1. The theory of groups," establishing that every finite group is isomorphic to a subgroup of the symmetric group acting on its own elements via the regular representation.^[8] This proof formalized the embedding by constructing explicit permutations corresponding to group multiplications, bridging concrete substitution groups with abstract structures.^[6] In the 1870s, Camille Jordan made pivotal contributions to permutation group theory, which underpinned and solidified the isomorphism central to Cayley's result. His 1870 treatise Traité des substitutions et des équations algébriques systematically classified substitution groups, introduced key concepts like primitive and imprimitive groups, and proved results on their structure, including the invariance of composition factors in series of subgroups (the Jordan-Hölder theorem).^[9] These advancements demonstrated how arbitrary finite groups could be realized as transitive permutation groups, reinforcing the theorem's implications for embedding in symmetric groups.^[10] During the 1870s and 1880s, Jordan, along with contemporaries like Otto Hölder, further entrenched the theorem by extending analyses of permutation representations to broader group classifications. Hölder's 1889 extension of the Jordan-Hölder theorem to abstract groups emphasized the role of symmetric group subgroups in revealing invariant structural properties, such as simple factors in composition series.^[6] This work collectively affirmed the universality of the isomorphism for finite groups, moving beyond ad hoc examples to general principles. The late 19th century saw a profound evolution from reliance on concrete permutation examples—rooted in solving polynomial equations—to a fully abstract group theory, where Cayley's embedding served as a foundational tool. Walther von Dyck's papers in 1882 and 1883 introduced the modern definition of groups via generators and relations, enabling proofs of properties like isomorphism to permutation subgroups without presupposing a faithful action on a set.^[6] This abstraction, building directly on Cayley's and Jordan's insights, laid the groundwork for 20th-century developments in group theory.

Mathematical Background

Basic Group Theory

A group is a fundamental algebraic structure consisting of a nonempty set G equipped with a binary operation \cdot: G \times G \to G, often denoted multiplicatively, that satisfies four axioms: closure, under which the product of any two elements in G remains in G; associativity, meaning (a \cdot b) \cdot c = a \cdot (b \cdot c) for all a, b, c \in G; the existence of an identity element e \in G such that a \cdot e = e \cdot a = a for all a \in G; and the existence of inverses, where for each a \in G, there is an element a^{-1} \in G satisfying a \cdot a^{-1} = a^{-1} \cdot a = e.^[11] The operation is not required to be commutative unless specified otherwise, distinguishing groups from abelian groups where a \cdot b = b \cdot a holds for all elements.^[11] Classic examples illustrate these properties. The set of integers \mathbb{Z} under addition forms an infinite abelian group, with the identity element 0 and the inverse of n being -n, satisfying all axioms since addition is closed, associative, and commutative on \mathbb{Z}.^[11] Another example is the symmetric group S_n, the set of all bijections (permutations) from a finite set of n elements to itself, under function composition; it has order n!, the identity is the identity permutation, and every permutation has an inverse, making S_n a non-abelian group for n \geq 3.^[11] Permutation groups like S_n represent a key class of finite groups central to symmetry studies.^[11] Homomorphisms provide a way to map between groups while preserving their structure. A group homomorphism is a function \phi: G \to H between groups (G, \cdot) and (H, \star) such that \phi(a \cdot b) = \phi(a) \star \phi(b) for all a, b \in G, ensuring the image of the operation in G corresponds to the operation in H.^[11] An isomorphism is a bijective homomorphism, meaning it is both injective and surjective, and thus has an inverse that is also a homomorphism; isomorphic groups are essentially the same up to relabeling of elements, capturing structural equivalence.^[11] These mappings are essential for comparing groups and understanding their properties without examining every element directly.^[11]

Permutation Groups and Actions

In group theory, the symmetric group on a set X, often denoted \mathrm{Sym}(X) or S_X, is the group consisting of all bijections from X to itself, with the group operation defined by function composition.^[12] The identity element is the identity function on X, and every bijection has an inverse that is also a bijection, ensuring the structure forms a group.^[13] When X is finite with |X| = n, \mathrm{Sym}(X) is denoted S_n and has order n!./04:_Families_of_Groups/4.03:_Symmetric_Groups) A permutation group is defined as a subgroup of \mathrm{Sym}(X) for some set X./06:_Permutation_and_Dihedral_Groups/6.01:_Introduction_to_Permutation_Groups) This perspective views such groups as collections of permutations of X that are closed under composition and include the identity permutation.^[14] In the context of Cayley's theorem, considering the symmetric group \mathrm{Sym}(G) on the underlying set of a group G provides a framework for embedding arbitrary groups into permutation groups.^[15] A group action of a group G on a set X is a homomorphism \phi: G \to \mathrm{Sym}(X), which assigns to each element g \in G a permutation \phi(g) \in \mathrm{Sym}(X) such that \phi(gh) = \phi(g) \circ \phi(h) for all g, h \in G.^[16] Equivalently, it can be described as a map G \times X \to X, denoted (g, x) \mapsto g \cdot x, satisfying e \cdot x = x and (gh) \cdot x = g \cdot (h \cdot x) for the identity e \in G.^[17] A key example is the action of G on a set via left multiplication, where group elements permute points according to the group's own operation.^[16] The action is called faithful if the homomorphism \phi is injective, meaning distinct elements of G induce distinct permutations on X.^[18] In this case, G is isomorphic to its image \phi(G), which is a subgroup of \mathrm{Sym}(X), thereby embedding G as a permutation group.^[19] Faithful actions thus establish a direct connection between abstract groups and concrete permutation representations, central to results like Cayley's theorem.^[16]

Statement and Proof

Formal Statement

Cayley's theorem asserts that every group G is isomorphic to a subgroup of the symmetric group \Sym(G), which consists of all bijections from the set G to itself under composition.^[20] A corollary of the theorem states that if G is a finite group of order n, then G is isomorphic to a subgroup of S_n, the symmetric group on n elements, where |\Sym(G)| = n!.^[20] This embedding holds for both finite and infinite groups, with the construction relying on the group's action on itself.^[21]

Standard Proof via Regular Action

The standard proof of Cayley's theorem constructs an embedding of a group G into the symmetric group \mathrm{Sym}(G) on the underlying set of G using the regular action of G on itself by left multiplication. This action defines a map \rho: G \to \mathrm{Sym}(G) given by \rho_g(h) = g h for all g, h \in G.^[22] Each \rho_g is a bijection on G, as left multiplication by a fixed group element is invertible with inverse given by left multiplication by g^{-1}.^[22] To establish that \rho is a group homomorphism, consider the composition of permutations. For any g, k \in G and h' \in G,

(\rho_g \circ \rho_k)(h') = \rho_g(\rho_k(h')) = \rho_g(k h') = g (k h') = (g k) h' = \rho_{g k}(h').

Thus, \rho_g \circ \rho_k = \rho_{g k}, confirming that \rho(gh) = \rho_g \circ \rho_h for all g, h \in G.^[22]^[16] The homomorphism \rho is injective because its kernel is trivial. Suppose \rho_g is the identity permutation on G, so \rho_g(h) = h for all h \in G. In particular, taking h = e (the identity element) yields g e = g = e. Therefore, \ker \rho = \{e\}, implying \rho is injective and G \cong \rho(G) \leq \mathrm{Sym}(G).^[22] This completes the proof, showing that every group is isomorphic to a subgroup of a symmetric group.^[22]

Alternative Proof Approaches

One alternative approach to proving Cayley's theorem employs the right-regular action of the group on itself, contrasting with the standard left-regular action. For a group G, define the map \rho_g: G \to G by \rho_g(h) = h g^{-1} for all h \in G. This map is a bijection because it has an inverse \rho_{g^{-1}}, as \rho_g \circ \rho_{g^{-1}} = \mathrm{id}_G = \rho_{g^{-1}} \circ \rho_g. The collection \{\rho_g \mid g \in G\} forms a subgroup of the symmetric group \mathrm{Sym}(G) under composition, since \rho_g \circ \rho_k = \rho_{gk} for all g, k \in G, and the map g \mapsto \rho_g is a group homomorphism from G to \mathrm{Sym}(G). This homomorphism is injective (hence faithful), as its kernel is trivial: if \rho_g = \mathrm{id}_G, then h g^{-1} = h for all h \in G, implying g = e. The right-regular representation yields a subgroup isomorphic to G, similar to the left-regular case. The two representations are isomorphic as abstract groups, though their images in \mathrm{Sym}(G) may differ. For finite groups, another variant presents the proof through the Cayley multiplication table, providing an intuitive embedding into the symmetric group. Label the elements of a finite group G with |G| = n as g_1 = e, g_2, \dots, g_n. The Cayley table has rows and columns indexed by these elements, with entry (i,j) given by g_i g_j = g_k for some k. Each row i corresponds to a permutation \sigma_i \in S_n that maps the column index j to k, i.e., \sigma_i(j) = k where g_i g_j = g_k. These permutations \{\sigma_i \mid 1 \leq i \leq n\} form a subgroup of S_n isomorphic to G, as the map sending g_i to \sigma_i is a homomorphism (composition of row permutations mirrors group multiplication) and injective (distinct rows differ, reflecting distinct left multiplications). This approach visually embeds G by viewing the table's rows as permutations.^[23] An abstract perspective on the theorem emphasizes the existence of a faithful permutation representation without detailing a specific action like the regular one. In general, a group action of G on a set X induces a homomorphism \phi: G \to \mathrm{Sym}(X) by \phi(g)(\xi) = g \cdot \xi for \xi \in X. The action is faithful if \phi is injective, meaning only the identity fixes all points. Cayley's theorem follows from the existence of such a faithful action on X = G (via any transitive free action), yielding an embedding into \mathrm{Sym}(G); the proof reduces to verifying that the induced \phi has trivial kernel, as non-identity elements move some points. This framework highlights the theorem as a consequence of faithful actions in group theory, applicable beyond the regular case.^[3]

The Regular Representation

Definition and Construction

The regular representation of a group G arises from its action on itself by left multiplication, providing an explicit construction for the embedding of G into a symmetric group as described in Cayley's theorem. This yields a permutation representation \rho: G \to \Sym(G), where \Sym(G) is the symmetric group on the set G, defined by \rho(h)(g) = h g for all h, g \in G.^[24] For a finite group G of order n, the regular representation can be viewed as a linear representation \rho: G \to \GL(n, \mathbb{C}). Let V = \mathbb{C}G be the complex vector space of dimension n with ordered basis \{e_g \mid g \in G\}. The map \rho is defined by its action on basis vectors: \rho_h(e_g) = e_{h g} for all h, g \in G, extended linearly to all of V.^[25] With respect to this basis, the matrix of \rho_h is the n \times n permutation matrix whose (i,j)-entry is 1 if the i-th basis vector is e_{h g} and the j-th is e_g for some g \in G, and 0 otherwise. This corresponds precisely to the permutation matrix realizing the left multiplication by h in the permutation representation.^[24] While the permutation representation \rho: G \to \Sym(G) acts solely by permuting the set G, the linear regular representation acts on the vector space V over \mathbb{C}, enabling the application of linear algebraic tools such as traces and inner products in subsequent analyses.^[25]

Key Properties

The regular action of a finite group G on itself by left multiplication, defined by g \cdot h = gh for g, h \in G, is transitive, meaning it has a single orbit consisting of all elements of G.^[26] This transitivity follows from the fact that for any g, h \in G, there exists a unique k = gh^{-1} \in G such that k \cdot h = g, ensuring every element can be reached from any other via the group action.^[27] As a permutation representation, the regular representation of G has degree |G|, corresponding to the action on the |G| elements of G itself.^[28] Over the complex numbers \mathbb{C}, the regular representation decomposes as a direct sum of all irreducible representations of G, where each irreducible representation \rho appears with multiplicity equal to its dimension \dim(\rho).^[28] In particular, the trivial representation appears exactly once in this decomposition, as its dimension is 1.^[29] For abelian groups G, all irreducible representations over \mathbb{C} are 1-dimensional, corresponding to the characters of G. In this case, the regular representation is the direct sum of all distinct 1-dimensional irreducible representations, each appearing with multiplicity 1.^[30] This decomposition reflects the fact that the character table of an abelian group has |G| distinct rows, each giving a unique irreducible character.^[31]

Relation to the Theorem

The regular representation \rho: G \to \Sym(G) defined by left multiplication, where \rho(g) maps each element h \in G to gh, yields an explicit faithful action of G on itself, thereby providing the isomorphism G \cong \rho(G) \leq \Sym(G) that realizes Cayley's theorem.^[3] This embedding demonstrates that every group is isomorphic to a subgroup of the symmetric group on its underlying set, with the image \rho(G) consisting precisely of those permutations induced by the group's own multiplication table.^[25] All regular representations of a given group G in \Sym(G) are unique up to conjugation: any two subgroups of \Sym(G) isomorphic to G via regular actions are conjugate to one another, as an equivariant bijection between the acted sets induces the conjugating element.^[32] The theorem and its regular representation extend to infinite groups, where the embedding remains faithful, but \Sym(G) becomes the uncountable group of all bijections on G (for countable infinite G), contrasting with the countable nature of G itself.^[33]

Examples and Applications

Concrete Group Examples

Cayley's theorem embeds the cyclic group \mathbb{Z}/3\mathbb{Z} of order 3 into the symmetric group S_3. Label the elements as \{0, 1, 2\} with addition modulo 3. The regular representation assigns to each element k \in \mathbb{Z}/3\mathbb{Z} the permutation \rho_k that acts by left addition: \rho_k(m) = m + k \pmod{3}. Thus, \rho_0 is the identity permutation, \rho_1 = (0\ 1\ 2), and \rho_2 = (0\ 2\ 1).^[34] The Klein four-group V_4, isomorphic to \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, consists of the identity e and three elements a, b, c satisfying a^2 = b^2 = c^2 = e and ab = ba = c. To apply the regular representation, label the elements as $1 = e, $2 = a, $3 = b, $4 = c. The corresponding permutations in S_4 are \sigma_a = (1\ 2)(3\ 4), \sigma_b = (1\ 3)(2\ 4), and \sigma_c = (1\ 4)(2\ 3), with \sigma_e the identity. This yields an embedding of V_4 as the subgroup of S_4 generated by these double transpositions.^[35] For the symmetric group S_3 itself, which has order 6, the regular representation provides an embedding into S_6. Here, S_3 acts on its own elements \{e, (1\ 2), (1\ 3), (2\ 3), (1\ 2\ 3), (1\ 3\ 2)\} by left multiplication, where each \sigma \in S_3 permutes the six positions according to \sigma \cdot \tau = \sigma \tau for \tau \in S_3. This faithful action realizes S_3 as a transitive subgroup of S_6.^[36]

Broader Implications and Uses

Cayley's theorem provides a foundational embedding of any finite group into the symmetric group, enabling the development of algorithms in computational group theory that treat abstract groups as permutation groups for efficient computation. This permutation representation allows the application of specialized algorithms, such as the Schreier-Sims algorithm, to recognize and analyze group structures by constructing a base and strong generating set, which facilitates tasks like determining subgroup lattices and testing membership with polynomial-time complexity in the degree of the permutation representation.^[37]^[38] In combinatorics, the theorem's construction of the regular action as a permutation representation underpins the use of tools like Burnside's lemma for counting distinct configurations under group symmetries. By viewing group actions on sets through their isomorphic embedding in the symmetric group, one can compute the average number of fixed points to enumerate orbits, such as the number of distinct colorings of a graph up to automorphism, thereby providing a systematic method for solving enumeration problems in symmetric structures.^[39]^[40] In physics, particularly quantum mechanics, Cayley's theorem facilitates the representation of symmetry groups—such as those describing particle exchanges or molecular configurations—as permutation groups, which is essential for computational simulations. This embedding allows for the exploitation of permutation symmetries in many-body systems to reduce the dimensionality of Hilbert spaces and accelerate numerical methods, for instance, in modeling quantum states of identical particles where antisymmetrization or symmetrization corresponds to irreducible representations of the symmetric group.^[41]^[42]

Generalizations

Extensions to Other Algebraic Structures

Cayley's theorem generalizes to semigroups by asserting that every finite semigroup S embeds as a subsemigroup of the full transformation monoid T_S on the set S. The embedding is achieved through the right regular representation \rho: S \to T_S, defined by \rho(s)(x) = x \cdot s for all x \in S. This map is a semigroup homomorphism because \rho(s \cdot t)(x) = x \cdot (s \cdot t) = (x \cdot s) \cdot t = \rho(s)(\rho(t)(x)), and for finite semigroups, it provides the desired embedding into the monoid of all self-maps under composition.^[43] For monoids, the analogue holds directly without adjoining an identity, as every monoid M embeds as a submonoid of the full transformation monoid T_M on M. The construction uses the regular representation \lambda: M \to T_M, given by \lambda(m)(n) = m \cdot n for m, n \in M. This is a monoid homomorphism preserving the identity via \lambda(e)(n) = e \cdot n = n, and it is injective because if \lambda(m) = \lambda(m'), then m = m \cdot e = m' \cdot e = m'. Thus, monoids inherit a faithful functional representation analogous to groups.^[44] Extensions to rings involve representations via endomorphisms of their additive structure. Specifically, every associative unital ring R embeds as a subring of the endomorphism ring \mathrm{End}_\mathbb{Z}(R^+), where R^+ denotes the additive abelian group of R. The embedding \phi: R \to \mathrm{End}_\mathbb{Z}(R^+) sends each r \in R to the endomorphism given by left multiplication by r, i.e., \phi(r)(x) = r \cdot x for x \in R. This preserves the ring operations: additivity follows from \phi(r + s)(x) = (r + s) x = r x + s x = \phi(r)(x) + \phi(s)(x), and multiplicativity from \phi(r s)(x) = (r s) x = r (s x) = \phi(r)(\phi(s)(x)); injectivity holds since if \phi(r) = 0, then r \cdot 1 = 0, so r = 0. Similar endomorphism representations apply to modules, where an R-module M can be viewed through actions on its underlying abelian group, embedding into appropriate endomorphism rings to capture the module structure.^[45]

Modern Interpretations

In category theory, Cayley's theorem admits an elegant reformulation concerning representability. Specifically, every group object in the category of sets, \mathbf{Set}, is corepresentable by its regular action, meaning the forgetful functor from the category of groups to \mathbf{Set} is corepresented by the regular representation \rho_G: G \to \mathrm{Sym}(G). This perspective arises as a consequence of the Yoneda lemma, which embeds the category of groups into the functor category \mathbf{Set}^{\mathbf{Grp}^{\mathrm{op}}}, where the regular action provides the corepresenting object. Here, the natural isomorphism \mathrm{Hom}_{\mathbf{Grp}}(G, H) \cong \mathrm{Nat}(h_G, U(H)) (with U the forgetful functor) underscores how the theorem captures the universal property of the regular permutation representation.^[46] For topological groups, Cayley's theorem extends to a continuous version, embedding any topological group G as a closed subgroup of the homeomorphism group \mathrm{Homeo}(X) of some locally compact Hausdorff space X, preserving the topology. This topological embedding ensures that the group operations remain continuous, providing a faithful realization of G within a concrete transformation group. In particular, for Lie groups—which are smooth manifolds equipped with compatible group operations—this yields continuous injections into the diffeomorphism group of the underlying manifold, facilitating the study of their geometric and analytic properties through permutation-like actions on spaces. This generalization highlights the theorem's robustness beyond discrete structures, enabling applications in differential geometry and representation theory of continuous groups. While Cayley's theorem applies uniformly to infinite groups via the explicit regular embedding into \mathrm{Sym}(G), its implications reveal non-constructive aspects and computational limitations, particularly for infinite cases. The embedding, though definable, does not yield effective algorithms for decision problems in general, as \mathrm{Sym}(G) for countable infinite G (isomorphic to \mathrm{Sym}(\mathbb{N})) contains subgroups with undecidable word problems, inherited from finitely presented groups with undecidable presentations via the theorem. For instance, the existence of such groups (e.g., Boone-Novikov groups) implies that subgroup membership or isomorphism testing within large symmetric groups can be undecidable, underscoring inherent limits in algorithmic group theory despite the theorem's existential power. This contrasts with finite cases, where permutation representations support computable verifications, but amplifies challenges in infinite settings by embedding undecidability into permutation models.^[47]

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Are all isomorphic simply transitive subgroups of $S_n$ conjugate?
Oct 9, 2020 · My guess is that the answer is either yes, or if not then there should be exactly two conjugacy classes of simply transitive subgroups ( ...Why do we care about two subgroups being conjugate?Enumerating all subgroups of the symmetric groupMore results from math.stackexchange.com
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Difficulty understanding Cayley theorem (group theory).
May 15, 2017 · The statement of the Cayley theorem is : Every group is isomorphic to a subgroup of some symmetric group. I am convinced if a given group is ...Missing: definition | Show results with:definition
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[PDF] Group Theory
(Cayley's Theorem.) Every group is isomorphic to a group of permutations. Proof. Let SG be the group of permutations of G. We will prove that ...
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[PDF] Abstract Algebra I
So Cayley's Theorem states that every group is isomorphic to a permutation group. The permutation representation afforded by left multiplication on the elements ...
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[PDF] 8 Groups of Permutations - UC Berkeley math
The multiplication table for S3 shows that S3 is not abelian. ... The map ϕ : G → SG defined by ∀x ∈ G, ϕ(x) = λx is called the left regular representation of G.
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(PDF) The Schreier-Sims algorithm - ResearchGate
Given an arbitrary generating set for a permutation group, the Schreier-Sims algorithm calculates a base and strong generating set. We describe ...
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[PDF] 1 Computational Group Theory 2 Groups given as Cayley Tables or ...
Oct 28, 2009 · Schreier-Sims Algorithm: from S, constructs a strong generating set, which consists of: – A sequence T1 ⊃ T2 ⊃···⊃ Tm ⊃ Tm+1 = ∅ of ...
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[PDF] Combinatorics: The Art of Counting - Michigan State University
Enumerative combinatorics has seen an explosive growth over the last 50 years. The purpose of this text is to give a gentle introduction to this exciting ...<|separator|>
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[PDF] Analysis and Applications of Burnside's Lemma - MIT Mathematics
May 17, 2018 · Abstract. Burnside's Lemma, also referred to as Cauchy-Frobenius Theorem, is a result of group theory that is used to count distinct objects.
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[PDF] Modeling Quantum Behavior in the Framework of Permutation Groups
Combining methods of computational group theory with Monte Carlo simulation we study a model based on representations of permutation groups. 1 Introduction.
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[PDF] 17 Permutation Groups and Many-Electron States
Permutation groups, also known as symmetric groups, are used to classify symmetry in many-electron states, arising from electron interchanges, and classify ...<|control11|><|separator|>
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[PDF] Analysis of Conctruction Cayley's Theorem on Groups, Semigroups ...
Jul 1, 2025 · Cayley's theorem discusses the embedding operations between algebraic structures which are carried out by showing that there is an injective ...
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[2406.19294] Short presentations for transformation monoids - arXiv
Jun 27, 2024 · Due to the theorems of Cayley and Vagner-Preston, the full transformation monoids and the symmetric inverse monoids play analogous roles in the ...
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Why is there no Cayley's Theorem for rings? - MathOverflow
Jun 12, 2010 · One interpretation of Cayley's theorem is that it gives you a cofinal sequence of finite groups, with respect to injective group homomorphisms, ...Smallest permutation representation of a finite group?2 Possible Generalizations of Cayley's Theorem?More results from mathoverflow.netMissing: citation | Show results with:citation
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[PDF] Category theory Supplemental notes 1
Feb 10, 2017 · A universal property is a uniform description of morphisms into or out of an object, and it characterizes an object uniquely up to isomorphism.
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[PDF] Geometric Group Theory - Clara Löh - Universität Regensburg
Jun 9, 2022 · Proposition 1.1.9 (Cayley's theorem). Every group is ... In fact, the proof of this theorem is the lion's share of the proof of the.