Abstract
Let <img src=image/13491971_01.gif> be a finite group. The probability that two selected elements <img src=image/13491971_02.gif> from <img src=image/13491971_03.gif> and <img src=image/13491971_04.gif> from <img src=image/13491971_01.gif> are chosen at random in a way that the greatest common divisor also known as gcd, of the order of <img src=image/13491971_02.gif> and <img src=image/13491971_04.gif>, which is equal to one, is called as the relative coprime probability. Meanwhile, another definition states that the vertices or nodes are the elements of a group and two distinct vertices or nodes are adjacent if and only if their orders are coprime and any of them is in the subgroup of the group and this is called as the relative coprime graph. This research focuses on determining the relative coprime probability and graph for cyclic subgroups of some nonabelian groups of small order and their associated graph properties by referring to the definitions and theorems given by previous researchers. Besides, various results of the relative coprime probability for nonabelian groups of small order are obtained. As for the relative coprime graph, the result shows that the domination number for each group is one whereas the number of edges and the independence number for each group vary. Types of graphs that can be formed are either star graph, planar graph or complete <img src=image/13491971_05.gif> subgraph depending on the order of the subgroup of a group.
Highlights
In the study of the extension of coprime graph, Williams [1] was the first person to introduce the prime graph in 1981, which later, this study has attracted the attention of many researchers until they began to extend the prime graph to the coprime graph and it is defined in Definition 1 below.Definition 1: [2] Coprime Graph of a GroupA graph whose vertices are elements of G and there are edges between two distinct vertices x and y if and only if ( x, y ) = 1 and it is called as a coprime graph ofG and is denoted as ΓG
The author determined the types and some properties of the graph and the results are stated in Section 2 since the Relative Coprime Probability and Graph for Some Nonabelian Groups of Small Order and
The results showed that the coprime probability of dihedral groups or p-groups are the same when G has the same order
Summary
In the study of the extension of coprime graph, Williams [1] was the first person to introduce the prime graph in 1981, which later, this study has attracted the attention of many researchers until they began to extend the prime graph to the coprime graph and it is defined in Definition 1 below. The author determined the types and some properties of the graph and the results are stated in Section 2 since the Relative Coprime Probability and Graph for Some Nonabelian Groups of Small Order and. Zulkifli and Mohd Ali in [4] and [5] continued the research of coprime graph done by Ma et al [2] but the scope of group was mainly on the nonabelian metabelian groups of order less than 24 and order 24, respectively Both papers were targeted on finding the types of graphs, the domination number, the independence number, and the number of edges that can be obtained within a group. In this paper, research on the coprime probability is continued further by introducing and determining the relative coprime probability for cyclic subgroups of some nonabelian groups of small order together with their graphs. The main results of this research are obtained and discussed in the third part of this paper and the overall research is summarized in the Conclusion section
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