scholarly journals SUBSTITUTION CHARACTERISTICS OF FACTORIAL SETS AND CRITERIA FOR CHOOSING THE SINGLE SUBSTITUTIONS

2020 ◽  
Vol 62 (4) ◽  
pp. 109-117
Author(s):  
Inna A. Martynova ◽  
Keyword(s):  
Permutation Groups ◽  
Group Operation ◽  
Cyclic Groups ◽  
Basic Concepts ◽  
One To One

The substitution and permutation function, which are presented in the article as elements of a number of factorial sets, are the key functions of cryptographic systems that provide diffusion and mixing of information. A new scale of notation is proposed while analyzing this problem. This is the notation scale of a number of factorial sets. This scale of notation helps to index the elements of a number of factorial sets and establish a one-to-one correspondence between the number and a specific type of substitution. This allows analyzing substitutions characteristics systematically. This paper presents the basic concepts of a number of the factorial sets. It is noted that the permutations of the factorial sets form symmetric permutation groups, and specific permutations (when raised to a power) form cyclic groups. The group axioms are fulfilled for the permutations of a number of factorial sets. Also, the definition domain, the group operation of multiplication, and identical and inverse substitutions are given for them. The number of independent cycles, decrement, inverse, parity and sign are common characteristics of the substitutions of a number of factorial sets. The criteria for choosing single substitutions with the best characteristics are proposed.

scholarly journals On Primitive Extensions of Rank 3 of Symmetric Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Permutation Group ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Minimal Set ◽  
Finite Set ◽  
Rank 2 ◽  
One To One

1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i))τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup GΔ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of GΔ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

scholarly journals A Study on Finite Abelian Groups: Sylow’s Theorems Based

2017 ◽  
Vol 1 (3) ◽  
Author(s):  
Amaira Moaitiq Mohammed Al-Johani
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Commutative Rings ◽  
Abstract Algebra ◽  
Vector Spaces ◽  
Finite Abelian Groups ◽  
Study Objective ◽  
Group Operation ◽  
Basic Concepts

In abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms. Algebraic structures include groups, rings, fields, and lattices, etc. A group is an algebraic structure (????, ∗), which satisfies associative, identity and inverse laws. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutatively. The concept of an Abelian group is one of the first concepts encountered in abstract algebra, from which many other basic concepts, such as rings, commutative rings, modules and vector spaces are developed. This study sheds the light on the structure of the finite abelian groups, basis theorem, Sylow’s theorem and factoring finite abelian groups. In addition, it discusses some properties related to these groups. The researcher followed the exploratory and comparative approaches to achieve the study objective. The study has shown that the theory of Abelian groups is generally simpler than that of their non-abelian counter parts, and finite Abelian groups are very well understood.  

A bee on a point, a line, and a plane

1966 ◽  
Vol 13 (7) ◽  
pp. 549-552
Author(s):  
Francis T. Sganga
Keyword(s):  
Elementary School ◽  
Number Line ◽  
School Mathematics ◽  
Elementary School Mathematics ◽  
Close Study ◽  
Basic Concepts ◽  
One To One

Most of the new elementary school mathematics programs place a great deal of emphasis upon such basic concepts as the number line, one-to-one correspondence, and number pairs. While there is general agreement that these concepts are important, a close study of how these concepts are developed indicates that it would be quite helpful to teachers and their students if greater emphasis were placed upon the significance of these concepts.

scholarly journals Cyclic Permutation Groups that are Automorphism Groups of Graphs

2019 ◽  
Vol 35 (6) ◽  
pp. 1405-1432 ◽  
Author(s):  
Mariusz Grech ◽  
Andrzej Kisielewicz
Keyword(s):  
Automorphism Group ◽  
Permutation Group ◽  
Boolean Functions ◽  
Automorphism Groups ◽  
Cyclic Permutation ◽  
Permutation Groups ◽  
Symmetry Groups ◽  
Cyclic Groups

Abstract In this paper we establish conditions for a permutation group generated by a single permutation to be an automorphism group of a graph. This solves the so called concrete version of König’s problem for the case of cyclic groups. We establish also similar conditions for the symmetry groups of other related structures: digraphs, supergraphs, and boolean functions.

The effects of instruction on the stage placement of children in Piaget's seriation experiments

1964 ◽  
Vol 11 (1) ◽  
pp. 4-9 ◽  
Author(s):  
Arthur F. Coxford
Keyword(s):  
Cardinal Number ◽  
Ordinal Number ◽  
Jean Piaget ◽  
Basic Concepts ◽  
One To One

In his book, The Child's Conceplion of Number, Jean Piaget 1 stated that the concept of number has three basic aspects: cardinal number, ordinal number, and unit. He has given criteria for determining when a child understands each of the basic concepts. A child understands cardinal number when he is able' to construct a one-to-one correspondence between two sets of objects and to conserve this corrspondcnce when it is no longer perceptually obvious.

A Construction for Vertex-Transitive Graphs

1982 ◽  
Vol 34 (2) ◽  
pp. 307-318 ◽  
Author(s):  
Brian Alspach ◽  
T. D. Parsons
Keyword(s):  
Permutation Groups ◽  
General Strategy ◽  
Circulant Graphs ◽  
Cyclic Groups ◽  
Disjoint Cycles ◽  
The Family ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group Γ in the family all graphs G whose vertex–set is the orbit V of Γ and for which Γ ≦ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of cyclic groups 〈(0 1 … n – 1)〉 generated by cycles (0, 1 … n – 1) of length n, then the corresponding graphs are the n-vertex circulant graphs.In this paper we consider transitive permutation groups of degree mn generated by a “rotation” ρ which is a product of m disjoint cycles of length n and by a “twisted translation” t; such that τρτ–l = ρα for some α.

scholarly journals Permutation Polytopes of Cyclic Groups

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Barbara Baumeister ◽  
Christian Haase ◽  
Benjamin Nill ◽  
Andreas Paffenholz
Keyword(s):  
Large Class ◽  
Cyclic Permutation ◽  
Permutation Groups ◽  
Convex Hulls ◽  
Facet Structure ◽  
Cyclic Groups ◽  
Permutation Matrices ◽  
Il Y A ◽  
International Audience ◽  
Permutation Polytope

International audience We investigate the combinatorics and geometry of permutation polytopes associated to cyclic permutation groups, i.e., the convex hulls of cyclic groups of permutation matrices. In the situation that the generator of the group consists of at most two orbits, we can give a complete combinatorial description of the associated permutation polytope. In the case of three orbits the facet structure is already quite complex. For a large class of examples we show that there exist exponentially many facets. Nous ètudions les propriètès combinatoires et gèomètriques des polytopes de permutations pour des groupes cycliques. C'est à dire, donnè un groupe cyclique de matrices de permutations, nous considèrons son enveloppe convexe. Si le gènèrateur du groupe possède un ou deux orbites il y a une dèscription simple du polytope. Par contre, le cas de trois (ou plus) orbites est beaucoup plus compliquè. Pour une classe ample d'examples nous construisons un nombre exponentiel de faces de co-dimension un.

A Cognitive-Linguistic Intervention Program: Basic Concept Formation Level

1977 ◽  
Vol 8 (1) ◽  
pp. 23-32
Author(s):  
Gerald E. Chappell
Keyword(s):  
Basic Concept ◽  
Concept Formation ◽  
Intervention Program ◽  
Cognitive Linguistic ◽  
Basic Concepts ◽  
Linguistic Skills ◽  
Linguistic Intervention

Test-teach questioning is a strategy that can be used to help children develop basic concepts. It fosters the use of multisensory exploration and discovery in learning which leads to the development of cognitive-linguistic skills. This article outlines some of the theoretical bases for this approach and indicates possibilities for their applications in child-clinician transactions.

A Support Program

1994 ◽  
Vol 25 (2) ◽  
pp. 112-114 ◽  
Author(s):  
Henna Grunblatt ◽  
Lisa Daar
Keyword(s):  
Hearing Aids ◽  
School Counselor ◽  
Support Program ◽  
Educational Audiology ◽  
Part Program ◽  
One To One

A program for providing information to children who are deaf about their deafness and addressing common concerns about deafness is detailed. Developed by a school audiologist and the school counselor, this two-part program is geared for children from 3 years to 15 years of age. The first part is an educational audiology program consisting of varied informational classes conducted by the audiologist. Five topics are addressed in this part of the program, including basic audiology, hearing aids, FM systems, audiograms, and student concerns. The second part of the program consists of individualized counseling. This involves both one-to-one counseling sessions between a student and the school counselor, as well as conjoint sessions conducted—with the student’s permission—by both the audiologist and the school counselor.

Prekindergarteners' Performance on The Boehm Test of Basic Concepts

1988 ◽  
Vol 16 ◽  
pp. 137-141 ◽  
Author(s):  
Amelia Ouellette ◽  
Robert Casteel
Keyword(s):  
Basic Concepts
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