Cayley Table Research Articles

The Group Permanent Determines the Finite Abelian Group

Let $G$ be a finite abelian group of order $n$ and $\mathcal M_G$ the Cayley table of $G$. Let $\mathcal P(G)$ be the number of formally different monomials occurring in $\mathsf {per}(\mathcal M_G)$, the permanent of $\mathcal M_G$. In this paper, for any finite abelian groups $G$ and $H$, we prove the following characterization $$\mathcal P(G)=\mathcal P(H)\ \Leftrightarrow\ G\cong H.$$ It follows that the group permanent determines the finite abelian group, which partially answers an open question of Donovan, Johnson and Wanless. In fact, $\mathcal P(G)$ is closely related to zero-sum sequences over finite abelian groups and we shall prove the above characterization by studying a reciprocity of zero-sum sequences over finite abelian groups. As an application of our method, we show that $\mathcal P(G)>\mathcal P(C_n)$ for any non-cyclic abelian group $G$ of order $n$ and thereby answer an open problem of Panyushev.

Quandles with one nontrivial column

The axioms of a quandle imply that the columns of its Cayley table are permutations. This paper studies quandles with exactly one non-trivially permuted column. Their automorphism groups, quandle polynomials, (symmetric) cohomology groups, and [Formula: see text] quandles are studied. The quiver and cocycle invariant of links using these quandles are shown to relate to linking number.

Exploration of Symmetric Groups: Cayley Tables, Subgroup Analysis, and Real-World Applications in Card Tricks

This research delves into the extensive analysis of symmetric groups of various orders and degrees. We explore different representations of permutations, construct a Cayley table for the symmetric group of degree four, and systematically identify all its subgroups using Lagrange's theorem. An intriguing discovery unfolds as we demonstrate that the converse of the theorem countered by Sylow's theorem does not hold universally. Furthermore, we apply the concepts of permutations and product of disjoint cycles to address a real-world problem – the Card Trick game. This study amalgamates theoretical exploration with practical applications, showcasing the versatility and depth of symmetric group theory.

New algorithm for the analysis of nucleotide and amino acid evolutionary relationships based on Klein four-group

Phylogenetics is the study of ancestral relationships among biological species. Such sequence analyses are often represented as phylogenetic trees. The branching pattern of each tree and its topology reflect the evolutionary relatedness between analyzed sequences. We present a Klein four-group algorithm (K4A) for the evolutionary analysis of nucleotide and amino acid sequences. Klein four-group set of operators consists of: identity e (U), and three elements—a = transition (C), b = transversion (G) and c = transition-transversion or complementarity (A). We generated Klein four-group based distance matrices of: 1. Cayley table (CK4), 2. Table rows (K4R), 3. Table columns (K4C), and 4. Euclidean 2D distance (K4E). The performance of the matrices was tested on a dataset of RecA proteins in bacteria, eukaryotes (Rad51 homolog) and archaea (RadA homolog). RecA and its functional homologs are found in all species, and are essential for the repair and maintenance of DNA. Consequently, they represent a good model for the study of evolutionary relationship of protein and nucleotide sequences. The ancestral relationship between the sequences was correctly classified by all K4A matrices concerning general topology. All distance matrices exhibited small variations among species, and overall results of tree classification were in agreement with the general patterns obtained by standard BLOSUM and PAM substitution matrices. During the evolution of a code there is a phase of optimization of system rules, the ambiguity of a code is eliminated, and the system starts producing specific components. Klein four-group algorithm is consistent with the concept of ambiguity reduction. It also enables the use of different genetic code table variants optimized for particular transitions in evolution based on biological specificity.

PROOF OF ALGEBRAIC STRUCTURES (RINGS AND FIELDS) WITH JAVA PROGRAMMING

Many branches of algebraic structures, such as rings and fields, are difficult to comprehend and undesirable due to their abstract character. The testing of algebraic structures can be aided by a computer software application, which makes it simpler and more fun to learn algebraic structures. This program is expected to make algebraic structural proofing easier, faster, and more accurate than manual proof. Users and the software in the application are connected through the Cayley table. The Java programming is just used to demonstrate the algebraic structures of rings and fields. The application program's proof results for the subject showed correct results with a quick processing time when compared to manual processing.

Algebraic characterization of Ifa main divination codes

Ifa is an age long divination system vastly consulted across west Africa, the Caribbeans and the Americas. The system goes with the use of signature codes named Odu Ifa which are 256 in number consisting of 16 Oju Odu (major) and 240 Omo Odu (minor). We consider the algebraic structure of this divination system by writing the codes representing the main Odu as 4 × 2 matrices which are found to satisfy group axioms under addition mod 2. We construct the Cayley table that gives an abelian group which we dubbed “self-inversed group”. Other properties of group structure are outlined.

Redesigning the Serpent Algorithm by PA-Loop and Its Image Encryption Application

This article presents a cryptographic encryption standard whose model is based on Serpent presented by Eli Biham, Ross Anderson, and Lars Knudsen. The modification lies in the design of the Cipher, we have used power associative (PA) loop and group of permutations. The proposed mathematical structure is superior to Galois Field (GF) in terms of complexity and has the ability to create arbitrary randomness due to a larger key space. The proposed method is simple and speedy in terms of computations, meanwhile it affirms higher security and sensitivity. In contrast to GF, PA-loop are non-isomorphic and have several Cayley table representations. This validates the resistance to cryptanalytic attacks, particularly those targeting mathematical structures. This cryptographic scheme’s full description of encryption and decryption is measured and rigorously assessed to support its multimedia applications. The observed speed of this technique, which uses a key of 256 bits and a block size of 128 bits, is comparable to three-key triple-DES.

Learning algebraic structures: Preliminary investigations

In this paper, we employ techniques of machine learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether artificial intelligence (AI) can “learn” algebraic structures. Using finite groups and finite rings as a concrete playground, we find that questions such as identification of simple groups by “looking” at the Cayley table or correctly matching addition and multiplication tables for finite rings can, at least for structures of small size, be performed by the AI, even after having been trained only on small number of cases. These results are in tandem with recent investigations on whether AI can solve certain classes of problems in algebraic geometry.

Sudoku pair Latin squares based on groups

A group-based Sudoku pair Latin square is a Sudoku pair Latin square obtained by applying an isotopism to the Cayley table of a finite group. We introduce a strategy for producing group-based Sudoku pair Latin squares and we use it to show that (kp−1,p)-Sudoku pair Latin squares exist for each prime p and each positive integer k≠3.

Algorithm for the set of similarity criteria formation for a physical process in the class of homogeneous functions

The efficiency of application of linear programming methods to problems of the theory of similarity and dimensions is shown. A general algorithm for formation of the set of similarity criteria for a physical process in the class of homogeneous functions is proposed. The set of systems of linear algebraic equations is created using the combinatorial method and chain diagrams. Basic and free variables and their corresponding variants of dimensionless sets of independent arguments, which are taken as the main similarity criteria, are distinguished. The set of derived similarity criteria is found using the basic criteria and the Cayley table.

Completing Partial Transversals of Cayley Tables of Abelian Groups

In 2003 Grüttmüller proved that if $n\geqslant 3$ is odd, then a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $2$ is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $k$ is completable to a transversal if and only if $n$ is odd and either $n \in \{k, k + 1\}$ or $n \geqslant 3k - 1$. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when $k = 3$ and $n$ is prime. In this paper, we prove Grüttmüller’s conjecture for $k = 2$ and $k = 3$ by establishing a more general result for Cayley tables of Abelian groups of odd order.

Transversals, Near Transversals, and Diagonals in Iterated Groups and Quasigroups

Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube.
 We prove that if a group $G$ of order $n$ satisfies the Hall–Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated quasigroups.

The geometry of diagonal groups

Diagonal groups are one of the classes of finite primitive permutation groups occurring in the conclusion of the O'Nan-Scott theorem. Several of the other classes have been described as the automorphism groups of geometric or combinatorial structures such as affine spaces or Cartesian decompositions, but such structures for diagonal groups have not been studied. The main purpose of this paper is to describe and characterise such structures, which we call diagonal semilattices. Unlike the diagonal groups in the O'Nan-Scott theorem, which are defined over finite characteristically simple groups, our construction works over any group, finite or infinite. A diagonal semilattice depends on a dimension m and a group T. For m=2, it is a Latin square, the Cayley table of T, though in fact any Latin square satisfies our axioms. However, for m>=3, the group T emerges naturally and uniquely from the axioms. (The situation somewhat resembles projective geometry, where projective planes exist in profusion but higher-dimensional structures are coordinatised by an algebraic object, a division ring.) A diagonal semilattice is contained in the partition lattice on a set, and we provide an introduction to the calculus of partitions. Many of the concepts and constructions come from experimental design in statistics. We also determine when a diagonal group can be primitive, or quasiprimitive (these conditions are equivalent for diagonal groups). Associated with the diagonal semilattice is a graph, the diagonal graph, which has the same automorphism group except in four small cases with m<=3. The class of diagonal graphs includes some well-known families, Latin-square graphs and folded cubes. We obtain partial results on the chromatic number of a diagonal graph, and mention an application to synchronization.

FRACTAL MAGMAS AND PUBLIC-KEY CRYPTOGRAPHY

In this paper, we deal with magmas the simplest algebras with a single binary operation. The main result of our research is algorithms for generating chain of finite magmas based on the self-similarity principle of its Cayley tables. In this way the cardinality of a magmas domain is twice as large as the previous one for each magma in the chain, and its Cayley table has a block-like structure. As an example, we consider a cyclic semigroup of binary operations generated by a finite magmas operation with a low-cardinality domain, and a modify the Diffie-Hellman-Merkle key exchange protocol for this case.

Latin squares with maximal partial transversals of many lengths

A partial transversal T of a Latin square L is a set of entries of L in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any maximal partial transversal of a Latin square of order n has size at least ⌈n2⌉ and at most n. We say that a Latin square is omniversal if it possesses a maximal partial transversal of all plausible sizes and is near-omniversal if it possesses a maximal partial transversal of all plausible sizes except one.Evans (2019) showed that omniversal Latin squares of order n exist for any odd n≠3. By extending this result, we show that an omniversal Latin square of order n exists if and only if n∉{3,4} and n≢2(mod4). Furthermore, we show that near-omniversal Latin squares exist for all orders n≡2(mod4).Finally, we show that no non-trivial finite group has an omniversal Cayley table, and only 15 finite groups have a near-omniversal Cayley table. In fact, as n grows, Cayley tables of groups of order n miss a constant fraction of the plausible sizes of maximal partial transversals. In the course of proving this, we partially solve the following interesting problem in combinatorial group theory. Suppose that we have two finite subsets R,C⊆G of a group G such that |{rc:r∈R,c∈C}|=m. How large do |R| and |C| need to be (in terms of m) to be certain that R⊆xH and C⊆Hy for some subgroup H of order m in G, and x,y∈G?

An algorithm for checking the existence of subquasigroups

Quasigroup-based cryptoalgorithms are being actively studied in the framework of theoretic projects; besides that, a number of quasigroup-based algorithms took part in NIST contests for selection of cryptographic standards. From the viewpoint of security it is highly desirable to use quasigroups without proper subquasigroups (otherwise transformations can degrade). We propose algorithms that take a quasigroup specified by the Cayley table as the input and decide whether there exist proper subquasigroups or subquasigroups of the order at least 2. Temporal complexity of the algorithms is optimized at the cost of increased spatial complexity. We prove bounds on time and memory and analyze the efficiency of software implementations applied to quasigroups of a large order. The results were reported at the XVIII International Conference «Algebra, Number Theory and Discrete Geometry: modern problems, applications and problems of history».

Infinite Latin Squares: Neighbor Balance and Orthogonality

Regarding neighbor balance, we consider natural generalizations of $D$-complete Latin squares and Vatican squares from the finite to the infinite. We show that if $G$ is an infinite abelian group with $|G|$-many square elements, then it is possible to permute the rows and columns of the Cayley table to create an infinite Vatican square. We also construct a Vatican square of any given infinite order that is not obtainable by permuting the rows and columns of a Cayley table. Regarding orthogonality, we show that every infinite group $G$ has a set of $|G|$ mutually orthogonal orthomorphisms and hence there is a set of $|G|$ mutually orthogonal Latin squares based on $G$. We show that an infinite group $G$ with $|G|$-many square elements has a strong complete mapping; and, with some possible exceptions, infinite abelian groups have a strong complete mapping.

Space efficient representations of finite groups

The Cayley table representation of a group uses O(n2) words for a group of order n and answers multiplication queries in time O(1). It is interesting to ask if there is a o(n2) space representation of groups that still has O(1) query-time. We show that for any δ, 1/log⁡n≤δ≤1, there is an O(n1+δ/δ) space representation for groups of order n with O(1/δ) query-time. We also show that for Dedekind groups, simple groups and several group classes defined in terms of semidirect product, there are O(n) space representations with logarithmic query-time where n is the size of the group. Farzan and Munro (ISSAC'06) defined a model for group representation and gave a succinct data structure for abelian groups with constant query-time. They asked if their result can be extended to categorically larger group classes. We construct data structures in their model for Hamiltonian groups and some other classes of groups with constant query-time.

The algebraic structure on the neutrosophic triplet set

The notion of the neutrosophic triplet was introduced by Smarandache and Ali. This notion is based on the fundamental law of neutrosophy that for an idea X, we have neutral of X denoted as neut(X) and anti of X denoted as anti(X). This paper studied a neutrosophic triplet set which is a collection of all triple of three elements that satisfy certain properties with some binary operation. Also given some interesting properties related to them. Further, in this paper investigated that from the neutrosophic triplet group can construct a classical group under multiplicative operation for ℤ n , for some specific n. These neutrosophic triplet groups are built using only modulo integer 2p, with p is an odd prime or Cayley table.

All group‐based latin squares possess near transversals

AbstractIn a latin square of order n, a near transversal is a collection of n −1 cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.