26 research outputs found
The Decomposition of a Finitely Generated Module over Some Special Ring
This research aims to give the decompositions of a finitely generated module over some special ring, such as the principal ideal domain and Dedekind domain. One of the main problems with module theory is to analyze the objects of the module. This research was using a literature study on finitely generated modules topics from scientific journals, especially those related to the module theory. And by selective cases we find a pattern to build a conjecture or a hypothesis, by deductive proof, we prove the conjecture and state it as a theorem. The main result in this study is the decomposition of the finitely generated module is a direct sum of the torsion submodule and torsion-free submodule. Since the torsion-free module is always a free module over a principal ideal domain, then the torsion-free submodule is a free module. Last, we generalize the ring, from a principal ideal domain, to a Dedekind domain. We found then the torsion-free submodule became a projective module. Then the decomposition of the finitely generated module is a direct sum of the torsion submodule and the projective submodule. These results should help the researchers to analyze the objects on module theory.
Subgrup Non Trivial Dari Grup Dihedral
Grup dikatakan grup dihedral dengan order , adalah grup yang dibangun oleh dua elemen dengan sifat . Grup dihedral dinotasikan dengan . Sama halnya dengan grup yang lain, grup dihedral juga memiliki subgrup. Pada paper ini akan dibahas teorema-teorema yang berkaitan dengan subgrup dihedral, adapun salah satunya hasilnya dapat memperlihatkan jika prima maka subgrup-subgrup dibagi kedalam 2 macam yaitu subgrup yang mengandung rotasi dan subgrup yang mengandung refleksi sedangkan jika komposit maka subgrup-subgrupnya dibagi kedalam 3 macam subgrup yaitu subgrup yang mengandung rotasi, refleksi dan gabungannya
Ekivalensi Ideal Hampir Prima dan Ideal Prima pada Bilangan Bulat Gauss
Kriptografi adalah salah satu cabang ilmu matematika yang banyak digunakan pada sistem keamanan digital. Kriptografi itu sendiri berkaitan dengan bilangan bulat dan sifat-sifatnya, terutama bilangan prima. Lebih spesifik, beberapa algoritma penting seperti RSA, sangat bergantung pada faktorisasi prima dari bilangan bulat. Abstraksi bilangan prima diperkenalkan oleh Dedekind pada tahun 1871, dikenal dengan nama ideal prima. Ideal prima diperumum oleh Bhatwadekar pada tahun 2009 dan dinamakan ideal hampir prima. Paper ini akan membuktikan bahwa ideal hampir prima dan ideal prima di bilangan bulat Gasuss adalah ekivale
Some Special Graphs of Quaternion Group
Research on an algebraic structure represented in graph theory opens the way for new research in recent years. Several types of new graphs continue to be developed, such as coprime and non-coprime graphs. This article will represent the quaternion group in several graphs, such as coprime graphs, non-coprime graphs, commuting graphs, non-commuting graphs, and identity graphs. We obtained several theorems about unique graphs. One of the results is that non-coprime graphs from the quaternion group are complete and regular graphs
THE INTERSECTION GRAPH REPRESENTATION OF A DIHEDRAL GROUP WITH PRIME ORDER AND ITS NUMERICAL INVARIANTS
One of the concepts in mathematics that developing rapidly today is Graph Theory. The development of Graph Theory has been combined with Group Theory, that is by representing a group in a graph. The intersection graph from group , noted by , is a graph whose vertices are all non-trivial subgroups of group and two distinct vertices are adjacent in if and only if . In this research the intersection graph of a Dihedral group, we looking for the shapes and numerical invariants. The results obtained are if for , then has a subgraphs and subgraphs , the girth of the graph is 3, radius and diameter of the graph in a row is 2 and 3, and the chromatic number of the graph is  
The Intersection Graph of a Dihedral Group
The intersection graph of a finite group G is a graph (V,E) where V is a set of all non-trivial subgroups of G and E is a set of edges where two distinct subgroups H_i , H_j are said to be adjacent if and only if H_i \cap H_j \neq {e} . This study discusses the intersection graph of a dihedral group D_{2n} specifically the subgraph, degree of vertices, radius, diameter, girth, and domination number. From this study, we obtained that if n=p^2 then the intersection graph of D_{2n} is containing complete subgraph K_{p+2} and \gamma(\Gamma_{D_{2n}})=p.
Algoritma Dijkstra dan Algoritma Semut dalam Menyelesaikan Masalah Lintasan Terpendek (Studi Kasus Jaringan Transportasi Pariwisata di Pulau Lombok)
On daily life, we often travel from one place to another place another by considering the efficiency, time and cost of having the accuracy required in determining the shortest path. The determination of the shortest path will be a consideration in the decision to show the path that will be pursued. The results obtained also require speed and accuracy with the help of computers. On testing, except performed the shortest route search, the search process is also carried out comparisons between Dijkstra’s Algorithm and Ant Algorithm. Refers to a previous study by Izzat Pratama on 2011 is more focused on the development of tourism transport models with 2-dimensional , Pascal-based visualization technology and use Dijkstra's algorithm in the process of finding a route. The study that researcher developed more emphasis on comparison of the algorithms that is the Dijkstra Algorithm and Ant Algorithm. The results of this study form the shortest path between two specified nodes along the tracks and speed (running time) of both the algorithm that used. The Dijkstra’s Algorithm in solving the shortest path between a specific pair of nodes has a speed 0 second and Ant Algorithm takes much longer, which is an average 37 seconds. However of the Ant Algorithm, the resulting trajectory is more varied. Of the results obtained, it can be said Dijkstra Algorithm is more efficient in the problem of finding the shortest path in the case of transportation networks on tourism of Lombok Island
Analisis Penyelesaian Rubik 2×2 Menggunakan Grup Permutasi
Penelitian ini bertujuan menyelesaikan Rubik 2x2, dipandang dari sudut matematika, yaitu bidang aljabar, khususnya grup permutasi. Seperti diketahui, Rubik ini memiliki enam warna berbeda dan dalam keadaan akhir, setiap sisinya memiliki warna yang sama. Dalam penyelesaianya, rubik ini pertama diacak dengan menggunakan koleksi permutasi yang mengakibatkan terjadi pemetaan untuk setiap kubus-kubus kecil. Selanjutnya dianalisis kondisi dan kasusnya dan diselesaikan Rubik 2x2 ini menggunakan grup permutas
SOME PROPERTIES OF COPRIME GRAPH OF DIHEDRAL GROUP D_2n WHEN n IS A PRIME POWER
The Study of algebraic structures, especially on graphs theory, leads to anew topics of research in recent years. In this paper, the algebraic structures that will be represented by a coprime graph are the dihedral group and its subgroups. The coprime graph of a group G, denoted by \Gamma_D_2n is a graph whose vertices are elements of G and two distinct vertices a and b are adjacent if only if (|a,|b|)=1. Some properties of the coprime graph of a dihedral group D_2n are obtained. One of the results is if n is prime then \Gamma_D_2n is a complete bipartite graph. Moreover, if n is the power of prime then \Gamma_D_2n is a multipartite graph
The Power Graph of a Dihedral Group
Graph theory is one of the topics in mathematics that is quite interesting to study because it is applicable and can be combined with other mathematical topics such as group theory. The combination of graph theory and group theory is that graphs can be used to represent a group. An example of a graph is a power graph. A power graph of the group is defined as a graph whose vertex set is all elements of and two distinct vertices and are connected if and only if or for a positive integer x and y. In this study, the author discusses the power graph of the dihedral group The results obtained from this study are the power graph of the dihedral group where with prime numbers and an natural number is a graph consisting of two non-disjoint subgraphs, namely complete subgraphs and star subgraphs. And we find that its radius and diameter are 1 and 2