The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs

The conjugation degree on a set is the probability that an element of a group fixes a set, whereby the group action considered is conjugation. The conjugation degree on a set is a variation of the commutativity degree of a group, which is the probability that two randomly chosen elements in a group commute. In this research, the presentation of metacyclic p-groups where p is an odd prime is used. Meanwhile, the set considered is a set of an ordered pair of commuting elements in the metacyclic p-groups, where p is equal to three and five, satisfying certain conditions. The conjugation degree on the set is obtained by dividing the number of orbits with the size of the set. Hence, the results are obtained by finding the elements of the group that follow the conditions of the ordered set, followed by the computation of the number of orbits of the set. In the second part of this research, the obtained results of the conjugation degree on a set are then associated with graph theory. The corresponding orbit graph, generalized conjugacy class graph, generalized commuting graph and generalized non-commuting graph are determined where a union of complete and null graphs, one complete and null graphs, one complete and null graphs with one empty and null graphs are found. Accordingly, several properties of these graphs are obtained, which include the degree of the vertices, the clique number, the chromatic number, the independence number, the girth, as well as the diameter of the graph. Furthermore, some new graphs are introduced, namely the orderly set graph, the order class graph, the generalized co-prime order graph, and the generalized non co-prime order graph, which resulted in the finding of one complete or empty graphs, a union of two complete or one complete graphs, a union of complete and empty graphs and a complete or empty graphs. Finally, several algebraic properties of these graphs are determined

The conjugation degree on a set of metacyclic 3-groups and 5-groups with their related graphs

The conjugation degree on a set is the probability that an element of a group fixes a set, whereby the group action considered is conjugation. The conjugation degree on a set is a variation of the commutativity degree of a group, which is the probability that two randomly chosen elements in a group commute. In this research, the presentation of metacyclic p-groups where p is an odd prime is used. Meanwhile, the set considered is a set of an ordered pair of commuting elements in the metacyclic p-groups, where p is equal to three and five, satisfying certain conditions. The conjugation degree on the set is obtained by dividing the number of orbits with the size of the set. Hence, the results are obtained by finding the elements of the group that follow the conditions of the ordered set, followed by the computation of the number of orbits of the set. In the second part of this research, the obtained results of the conjugation degree on a set are then associated with graph theory. The corresponding orbit graph, generalized conjugacy class graph, generalized commuting graph and generalized non-commuting graph are determined where a union of complete and null graphs, one complete and null graphs, one complete and null graphs with one empty and null graphs are found. Accordingly, several properties of these graphs are obtained, which include the degree of the vertices, the clique number, the chromatic number, the independence number, the girth, as well as the diameter of the graph. Furthermore, some new graphs are introduced, namely the orderly set graph, the order class graph, the generalized co-prime order graph, and the generalized non co-prime order graph, which resulted in the finding of one complete or empty graphs, a union of two complete or one complete graphs, a union of complete and empty graphs and a complete or empty graphs. Finally, several algebraic properties of these graphs are determined

The conjugation degree on a set of metacyclic 3-groups

Research on commutativity degree has been done by many authors since 1965. The commutativity degree is defined as the probability that two randomly selected elements in a group commute. In this research, an extension of the commutativity degree called the probability that an element of a group fixes a set Ω is explored. The group G in our scope is metacyclic 3-group and the set Ω consists of a pair of distinct commuting elements in the group G in which their orders satisfy a certain condition. Meanwhile, the group action used in this research is conjugation. The probability that an element of G fixes a set Ω, defined as the conjugation degree on a set is computed using the number of conjugacy classes. The result turns out to be 7/8 or 1, depending on the orbit and the order of Ω

On the generalized commuting and non-commuting graphs for metacyclic 3-groups

Let be a metacyclic 3-group and let be a non-empty subset of such that . The generalized commuting and non-commuting graphs of a group is denoted by and respectively. The vertex set of the generalized commuting and non-commuting graphs are the non-central elements in the set such that where Two vertices in are joined by an edge if they commute, meanwhile, the vertices in are joined by an edge if they do not commute

A relation between tudung saji weaving patterns and group theory

Tudung saji is a traditional utensil used by the Malays to cover their food to be served. Tudung saji is woven with strands of dried leaves using a specific technique called triaxial weave, where the strands are plaited in three directions. Previously, a tool known as triaxial template had been created to represent the patterns of tudung saji weaving in a planar pattern. Based on this template, many beautiful and symmetrical patterns were successfully generated, creating some of the original patterns of tudung saji. These patterns are categorized according to the number of colours of the strands, from the basic 2-strand up to 6-strand template. The purpose of study is to find several finite groups to represent the triaxial weaving patterns on two dimensional templates, focussing only on the 2 and 3-strand templates. It is found that the symmetric group of two letters, 2 S and the cyclic group of order six, 6 C are isomorphic to the triaxial template of Flock of Pigeons and Sailboats patterns, respectively. These isomorphisms are determined by mapping the elements of the Flock of Pigeons and Sailboats onto the elements of the two groups. Using a software iMac Grapher, several graphs are generated based on the elements of the triaxial template patterns. Next, graph theory is used to analyze the properties of these graphs. The graphs are sorted by the numbers of strands, namely the graphs of block two, graphs of block three up to the graphs of block six. All such graphs are found to feature the characteristics of three types of graphs, namely a complete graph with three vertices, 3 K , a simple graph with six vertices, and an a cyclic graph. Lastly, this research reports on modifications to the template by adding extra colours to the framework strands and the insertion strands. This is done by using new colour ordering in addition to the same colour ordering for the 2-strand template. As a result, a new characteristic on the modified template has been found, namely the existence of different triaxial patterns in one template

Modelling of tudung saji weaving using elements in group theory

This paper describes a relation between the practice of Malay food cover weaving and mathematics. The cover, known as tudung saji, is woven using a specific technique called triaxial or hexagonal weave. Its tessellated parallelograms form an illusion of three dimensional cubes that are found interesting in mathematical studies of symmetrical patterns using group theory. Some of the properties of triaxial template patterns of order n3 are also discussed in this pape

The analysis of crystallographic symmetry types in finite groups

Undeniably, it is human nature to prefer objects which are considered beautiful. Most consider beautiful as perfection, hence they try to create objects which are perfectly balance in shape and patterns. This creates a whole different kind of art, the kind that requires an object to be symmetrical. This leads to the study of symmetrical objects and pattern. Even mathematicians and ethnomathematicians are very interested with the essence of symmetry. One of these studies were conducted on the Malay traditional triaxial weaving culture. The patterns derived from this technique are symmetrical and this allows for further research. In this paper, the 17 symmetry types in a plane, known as the wallpaper groups, are studied and discussed. The wallpaper groups will then be applied to the triaxial patterns of food cover in Malaysia

Study and design of supports and methods for provision of stability of mine workings and based on principle of injection strengthening of rock

Available from VNTIC / VNTIC - Scientific & Technical Information Centre of RussiaSIGLERURussian Federatio

The conjugation degree on a set of metacyclic 3-groups

Research on commutativity degree has been done by many authors since 1965. The commutativity degree is defined as the probability that two randomly selected elements in a group commute. In this research, an extension of the commutativity degree called the probability that an element of a group fixes a set Ωis explored. The group G in our scope is metacyclic 3-group and the set Ω consists of a pair of distinct commuting elements in the group G in which their orders satisfy a certain condition. Meanwhile, the group action used in this research is conjugation. The probability that an element of G fixes a set Ω, defined as the conjugation degree on a set is computed using the number of conjugacy classes. The result turns out to be 7/8 or 1, depending on the orbit and the order of Ω

Vicinity condition of online examination : does it affect the academic performances of pre-university students?

Physical attendance of students in a hall for examinations is no longer being practiced during the global pandemic of Coronavirus disease 2019 (Covid-19). Most universities, including Universiti Sultan Zainal Abidin (UniSZA) have adapted to online exams. This paper presents an analysis of online final examination performances of 251 students from the Science and Medicine Foundation Centre, UniSZA who completed foundation for 2019/2020 and 2020/2021 intake sessions. The objective of this research is to compare the final scores of Semester 2 for the students from both sessions, if there is a significant difference between academic performance by taking online examination at home for 2019/2020 intake session and at the university for 2020/2021 intake session based on the subjects of Mathematics II, Physics II, Chemistry II, Biology II and Information Technology II. The z-test was performed to compare mean scores for each subject for both intake sessions using Microsoft Excel worksheet. According to the mean value, students achieved higher score in Biology II, Chemistry II and Information Technology II by taking examination at hostel. Meanwhile, Physics II and Mathematics II subjects shows that the score are higher while taking examination at home. The p-value for each subject is computed, and the result is less than 0.025. The null hypothesis is then rejected