Cayley's Theorem Research Articles

A note on hamiltonicity conditions of the coprime and non-coprime graphs of a finite group

Let $G$ be a group. The coprime and non-coprime graphs of $G$ are introduced by Ma et al. (2014) and Mansoori et al. (2016), respectively, when $G$ is finite. By their definitions, which refer to coprime and non-coprime terms of two positive integers, those graphs must be related. We prove that they are closely related through their graph complement and preserve the isomorphism groups. Furthermore, according to Cayley's theorem, which states that any group $G$ is isomorphic to a subgroup of the symmetric group on $G$, it implies that the studies of the coprime and non-coprime graphs of any group $G$ (especially, when $G$ is finite) can actually be represented by the coprime and non-coprime graphs of any subgroup of the symmetric group on $G$. This encourages us to specifically study the hamiltonicity of both kinds of graphs associated with $G$ when $G$ is isomorphic to the symmetric group on $G$.

Endomorphisms and anti-endomorphisms of some finite groupoids

In this paper, we study anti-endomorphisms of some finite groupoids. Previously, special groupoids S(k,q) of order k(1+k) with a generating set of k elements were introduced. Previously, the element-by-element description of the monoid of all endomorphisms (in particular, automorphisms) of a given groupoid was studied. It was shown that every finite monoid is isomorphically embeddable in the monoid of all endomorphisms of a suitable groupoid S(k,q). In recent article, we give an element-by-element description for the set of all anti-endomorphisms of the groupoid S(k,q). We establish that, depending on the groupoid S(k,q), the set of all its anti-endomorphisms may be closed or not closed under the composition of mappings. For an element-by-element description of anti-endomorphisms, we study the action of an arbitrary anti-endomorphism on generating elements of a groupoid. With this approach, the anti-endomorphism will fall into one of three classes. Anti-endomorphisms from the two classes obtained will be endomorphisms of given groupoid. The remaining class of anti-endomorphisms, depending on the particular groupoid S(k,q), may either consist or not consist of endomorphisms. In this paper, we study endomorphisms of some finite groupoids G whose order satisfies some inequality. We construct some endomorphisms of such groupoids and show that every finite monoid is isomorphically embedded in the monoid of all endomorphisms of a suitable groupoid G. To prove this result, we essentially use a generalization of Cayley's theorem to the case of monoids (semigroups with identity).

Una Demostración del Teorema de Cayley para Grupos

Una Demostración del Teorema de Cayley para Grupos

New representaions of elements of skew linear recurrent sequences via trace function based on the noncommutative Hamilton - Cayley theorem

Пусть $p$ - простое число, $R=\mathrm{GR}(q^d,p^d)$ - кольцо Галуа мощности $q^d$ и характеристики $p^d$, где $q = p^r$, $S=\mathrm{GR}(q^{nd},p^d)$ - расширение степени $n$ кольца $R$, а $\mathrm{End}(_RS)$ - кольцо эндоморфизмов модуля $_RS$. Последовательность $v$ над $S$, удовлетворяющая закону рекурсии $$ \forall i\in\mathbb{N}_0\colon v(i+m)= \psi_{m-1}(v(i+m-1))+\ldots+\psi_0(v(i)), $$ $\psi_0,\ldots,\psi_{m-1 }\in \mathrm{End}(_RS)$, называется скрученной линейной рекуррентной последовательностью (ЛРП) над $S$; ее максимально возможный период равен $(q^{mn}-1)p^{d-1}$. С использованием функции след для представлений элементов скрученной ЛРП максимального периода доказано, что такая ЛРП линеаризуема, если коэффициенты в законе рекурсии попарно коммутируют.

A proof to direct sum decomposition for finite dimension of linear space

In higher algebra, Direct sum decomposition theorem for finite dimension of linear space is a classical result of linear transformation. This theorem has been widely used in many fields, such as algebra and mechanics and so on. In this paper, the proof is given by means of Hamilton Cayley theorem.

Actions of soft groups

The soft set theory proposed by Molodtsov is a recent mathematical approach for modeling uncertainty and vagueness. The main aim of this study is to introduce the concept of soft action by combining soft set theory with the action which is an important concept in dynamical systems theory. Moreover, different types of soft action are presented and some important characterizations are given. Finally, we define the concept of soft symmetric group and present the relation between the soft action and soft symmetric group, as a similar result to the classical Cayley's Theorem.

Cayley's Theorem and Adams completion

Cayley's Theorem and Adams completion

Reversible Monadic Computing

We extend categorical semantics of monadic programming to reversible computing, by considering monoidal closed dagger categories: the dagger gives reversibility, whereas closure gives higher-order expressivity. We demonstrate that Frobenius monads model the appropriate notion of coherence between the dagger and closure by reinforcing Cayley's theorem; by proving that effectful computations (Kleisli morphisms) are reversible precisely when the monad is Frobenius; by characterizing the largest reversible subcategory of Eilenberg–Moore algebras; and by identifying the latter algebras as measurements in our leading example of quantum computing. Strong Frobenius monads are characterized internally by Frobenius monoids.

Using Cayley's theorem to find the order of a power in a group

It is shown how Cayley's theorem can be used to prove the formula for the order of a power of an element of finite order in a group. Reasoning with disjoint cycles leads to a proof that depends on elementary number theory in some new ways, leading naturally to some new connections involving least common multiples, greatest common divisors and minima.

Symmetric Groups under multiset perspective

A multiset is a collection of objects in which repetition of elements is significant. In this paper an attempt to define a symmetric group under multiset context is presented and the analogous to Cayley's theorem is derived.

The number of spanning trees in an (r, s)‐semiregular graph and its line graph

AbstractFor a graph G, a “spanning tree” in G is a tree that has the same vertex set as G. The number of spanning trees in a graph (network) G, denoted by t(G), is an important invariant of the graph (network) with lots of decisive applications in many disciplines. In the article by Sato (Discrete Math. 2007, 307, 237), the number of spanning trees in an (r, s)‐semiregular graph and its line graph are obtained. In this article, we give short proofs for the formulas without using zeta functions. Furthermore, by applying the formula that enumerates the number of spanning trees in the line graph of an (r, s)‐semiregular graph, we give a new proof of Cayley's Theorem. © 2013 Wiley Periodicals, Inc.

Cayley's Theorem

The notation and terminology used in this paper have been introduced in the following papers: [3], [6], [4], [5], [10], [11], [7], [2], [1], [9], and [8]. In this paper X, Y denote sets, G denotes a group, and n denotes a natural number. Let us consider X. Note that ∅X,∅ is onto. Let us observe that every set which is permutational is also functional. Let us consider X. The functor permutationsX is defined as follows: (Def. 1) permutationsX = {f : f ranges over permutations of X}. Next we state three propositions: (1) For every set f such that f ∈ permutationsX holds f is a permutation of X. (2) permutationsX ⊆ XX . (3) permutations Seg n = the permutations of n. Let us consider X. One can verify that permutationsX is non empty and functional. Let X be a finite set. One can verify that permutationsX is finite. Next we state the proposition (4) permutations ∅ = 1. Let us consider X. The functor SymGroupX yields a strict constituted functions multiplicative magma and is defined by:

A Cayley theorem for distributive lattices

A Cayley theorem for distributive lattices

BINARY REPRESENTATIONS OF ALGEBRAS WITH AT MOST TWO BINARY OPERATIONS: A CAYLEY THEOREM FOR DISTRIBUTIVE LATTICES

The notion of binary representation of algebras with at most two binary operations is introduced in this paper, and the binary version of Cayley theorem for distributive lattices is given by hyperidentities. In particular, we get the binary version of Cayley theorem for DeMorgan and Boolean algebras.

Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: Beyond Poncelet porisms

The thirty years old programme of Griffiths and Harris of understanding higher-dimensional analogues of Poncelet-type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper. Starting with the observation of the billiard nature of some classical constructions and configurations, we construct the billiard algebra, that is a group structure on the set T of lines simultaneously tangent to d − 1 quadrics from a given confocal family in the d-dimensional Euclidean space. Using this tool, the related results of Reid, Donagi and Knörrer are further developed, realized and simplified. We derive a fundamental property of T: any two lines from this set can be obtained from each other by at most d − 1 billiard reflections at some quadrics from the confocal family. We introduce two hierarchies of notions: s-skew lines in T and s-weak Poncelet trajectories, s = − 1 , 0 , … , d − 2 . The interrelations between billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians developed in this paper enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results: Cayley's theorem, Weyr's theorem, Griffiths–Harris theorem and Darboux theorem.

Cayley's Theorem and Hopf Galois structures for semidirect products of cyclic groups

For G any finite group, the left and right regular representations λ, respectively ρ of G into Perm ( G ) map G into InHol ( G ) = ρ ( G ) ⋅ Inn ( G ) . We determine regular embeddings of G into InHol ( G ) modulo equivalence by conjugation in Hol ( G ) by automorphisms of G, for groups G that are semidirect products G = Z h ⋊ Z k of cyclic groups and have trivial centers. If h is the power of an odd prime p, then the number of equivalence classes of regular embeddings of G into InHol ( G ) is equal to twice the number of fixed-point free endomorphisms of G, and we determine that number. Each equivalence class of regular embeddings determines a Hopf Galois structure on a Galois extension of fields L / K with Galois group G. We show that if H 1 is the Hopf algebra that gives the standard non-classical Hopf Galois structure on L / K , then H 1 gives a different Hopf Galois structure on L / K for each fixed-point free endomorphism of G.

Arthur Cayley: mathematician laureate of the Victorian age

Arthur Cayley (1821-1895) was one of the most prolific and important mathematicians of the Victorian era. His influence still pervades modern mathematics, in group theory (Cayley's theorem), matrix algebra (the Cayley-Hamilton theorem), and invariant theory, where he made his most significant contributions. Yet Cayley's life has been overlooked by historians, who have lavished far more attention on lesser figures. and biographer Tony Crilly, the world's leading authority on Cayley, rectifies this oversight with the first definitive account of his life. Born in England, Cayley spent his childhood in St. Petersburg, where his father was a commercial agent. After returning to England in 1828, Cayley received a first-rate education. As an undergraduate at Trinity College in Cambridge, he was named Senior Wrangler, the top mathematics student of his year. After graduating, he found himself at the vanguard of the revolution in British mathematics which included William Rowan Hamilton, George Boole, and James Joseph Sylvester. At the same time, needing a reliable income, he trained for the bar and became a barrister at Lincoln's Inn in 1849. Though a successful lawyer, Cayley devoted all his free time to mathematics and confirmed his reputation as one of the era's leading minds with a procession of brilliant articles on key aspects in pure mathematics. Only after 1863, when he was appointed to the Sadleirian Chair at Cambridge, could he fully pursue mathematical investigations, and he continued to publish influential papers until his death. Comprehensive and elegantly composed, this biography makes clear the scope of Arthur Cayley's prodigious achievements, firmly enshrining him as the Mathematician Laureate of the Victorian Age.

Counting nilpotent endomorphisms

A variant of Prüfer's classical proof of Cayley's theorem on the enumeration of labelled trees counts the nilpotent self-maps of a pointed finite set. Essentially, the same argument can be used to establish the result of Fine and Herstein [Illinois J. Math. 2 (1958) 499–504] that the number of nilpotent n × n matrices over the finite field F q is q n ( n - 1 ) .

Immersions of Einstein-Riemann spacetimes infive-dimensional flat spaces

Gauge-theoretic constructs and Cartan's method of movingcoframe fields are used to obtain immersions of four-dimensionalEinstein-Riemann spacetimes in flat five-dimensional spaces. Ageneralization of Cayley's representation theorem (for matrix elements ofSO(n)) to the groups SO(2,3) and SO(1,4) leads to a choice of thefourimmersion parameters such that all salient geometric quantities areevaluated in terms of rational algebraic functions. After an analysis ofthe general immersion problem, considerations are restricted to classesof immersions for which the matrix of gauge curvature 2-forms is ofmaximal rank. All solutions of these immersion problems are shown to begenerated by vector spaces of 1-form fields, so that there is asuperposition principle at this level. These 1-form fields lead to familiesof associated metric and curvature tensor fields with specificcomposition laws. Several multiple-parameter families of immersions arecalculated explicitly. A spherically symmetric, deflation-inflation modelwith a central black hole is presented.

On \(SU(2)\;{ \times }\;\mathcal{S}_{n \geqslant 12} \) dual‐group tensorial sets and carrier spaces in the multiple invariant physics of multiquantum NMR. II. \(\;\{ \mathcal{S}_{12} \supset \cdot \cdot \cdot \supset [2]\mathcal{S}_2 \} \) pathways from Yamanouchi monomial reductions as [A]12 democratic Sn‐invariant labels

The transformational and Liouville carrier space (LCS) properties of dual tensorial bases for [A...]n NMR spin systems are considered within the quantum physics of (super)boson quasiparticles, where the (contracted) auxiliary labels of \(\;SU(2)\;{ \times }\;\mathcal{S}_n \) tensors are derived from the Sn‐based scalar invariants (SIs). Beyond both the {\(s_i^2 \)} (super)bosons mappings of pattern‐algebra [F.P. Temme, Physica A 198 (1993) 245] and the (outer) k‐rank based sub‐structure of LCS (e.g., in terms of S12 irreps [F.P. Temme, J. Math. Chem. 27 (2000) 111 (this issue)]), now (cf. Jucys recoupling) we consider the dual group physics role of the \({\widetilde {\mathcal{V}}} \) auxiliary terms of the (SI‐related) democratic sets for all the \( \geqslant \)4‐fold multispin systems, as obtained via the \(\mathcal{S}_{n - 1} \supset \cdot \cdot \cdot \supset ([2])\mathcal{S}_2 \) Yamanouchi–Gel'fand chains (YGCs). The simple reducibility of LCS derives from the explicit role of such auxiliary labels in dual mapping. The full monomial YGC reduction coefficient sets, and their related sum rule, are given here for \(SU(2)\;{ \times }\;\mathcal{S}_{12} \) tensorial sets and the distinctness of individual reduction pathways is demonstrated. Recent enumerative work on SIs [F.P. Temme, J. Magn. Reson. (2000) (to appear)] (extending [P.L. Corio, J. Magn. Reson. 134 (1998) 131]) gives expressions for the numbers of independent S12(S20) SIs for icosa‐(dodeca)hedral spin ensembles. The search for additional insight into multiquantum evolution (or coherence transfer) from the use of \(\left\{ {{\mathcal{T}}_{\left\{ {\widetilde{\mathcal{V}}} \right\}}^{kq} \left( {1_1 ,...,1_n ;\left[ {\widetilde\lambda } \right]} \right)} \right\}\) dual bases motivates this work – cf. that of [M.C. Carravetta et al., J. Magn. Reson. 134 (1998) 131; B.C. Sanctuary, Molecular Phys. 55 (1985) 1017]. Studies of SIs, and of the origins of Cayley's group embedding theorem, highlight the need to retain the Sn group in quantum physics involving spin ensembles, cf. Corio's O(3) viewpoint. A recent lattice‐point model of 12‐fold cage isotopomers, now for general multipartite forms, has demonstrated that universal mathematical determinacy (a property for which YGCs are especially noted) also prevails in \(SU(m)\;{ \times }\;\mathcal{S}_{12} \; \downarrow \mathcal{I}\) natural group embedding [F.P. Temme, Eur. Phys. J. 11(1999) 177].