Co-prime Probability for Nonabelian Metabelian Groups of Order Less than 24 and Their Related Graphs

Zulkifli, Nur Amalina; Ali, Nor Muhainiah Mohd

doi:10.11113/matematika.v35.n3.1130

Cited by 5 publications

(8 citation statements)

References 6 publications

(7 reference statements)

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“…Zulkifli and Mohd Ali [9] conducted an extensive study in this part which they determined the coprime probability for nonabelian metabelian groups of order less than 24. Here, it may be discovered that if G has the same order, then the coprime probability of nonabelian metabelian groups of order less than 24 are the same.…”

Section: Definition 2: [4] Noncoprime Graphmentioning

confidence: 99%

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Zulkifli¹,

Ali²

2022

Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)

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The study of coprime probabilities and graphs have its own uniqueness that produces a particular pattern according to its variabilities. Some obvious results can be seen from previous research where the domination number will always be equal to one and the types of graphs that can be formed are either star, planar or r-partite graph depending on certain cases. For the probability, the results vary according to the groups and certain cases need to be considered. The noncoprime graph has been introduced and it is defined as a graph associated to the group G with vertex set G\{e} such that it is possible that two separate vertices are adjacent when the orders are relatively noncoprime. However, in probability theory, the study of noncoprime probability of a group has not been introduced yet. Hence, a thorough study has been conducted where the goal of this research is to introduce a newly defined graph and probability which are the prime power noncoprime graph and prime power noncoprime probability of a group. The focus of this approach is that the greatest common divisor of the order of x and y, where x and y are in G, is equal to a power of prime number. In this paper, the scope of the group is mainly focused on some dihedral groups, quasi-dihedral groups, and some generalized quaternion groups. Some invariants, which are the diameter, girth, clique number, chromatic number, domination number, and independence number of prime power noncoprime graph are found. Additionally, the generalization of the prime power noncoprime probability are also obtained.

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Section: Definition 2: [4] Noncoprime Graphmentioning

confidence: 99%

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Zulkifli¹,

Ali²

2022

Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)

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“…In the paper, the author determined the types and some properties of the graph and the results are stated in Section 2 since the Their Associated Graph Properties results will be used throughout this research. Recently, 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.…”

Section: Definition 1: [2] Coprime Graph Of a Groupmentioning

confidence: 99%

“…and n is odd are discussed. Later, Zulkifli and Mohd Ali [4] made an extensive research on the coprime probability in which the scope of the group is on the nonabelian metabelian groups of order less than 24. The results showed that the coprime probability of dihedral groups or -groups p are the same when G has the same order.…”

Section: Definition 1: [2] Coprime Graph Of a Groupmentioning

confidence: 99%

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Relative Coprime Probability and Graph for Some Nonabelian Groups of Small Order and Their Associated Graph Properties

Zulkifli¹,

Ali²

2021

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“…The authors found the probability for p-groups and for some dihedral groups. Later in 2019, Zulkifli and Mohd Ali in [21] and [20] made an extensive research on the coprime probability in which the scope of the group is on the nonabelian metabelian groups of order less than 24 and order 24, respectively.…”

Section: Introductionmentioning

confidence: 99%

The Order Product Prime Probability and Commutativity Degree for Some Finite Groups

Ali¹,

Isah²,

Bello³

2022

MJMS

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Statistical approach often provides techniques in exploring more on the algebraic properties of finite groups. In this paper, two new group probabilities are defined in which the group elements having certain common characteristics are considered. Let G be a finite group, the order product prime probability is defined as the probability that two randomly selected elements x and y in G satisfy |x||y|=ps and the order product prime commutativity degree is the probability that elements satisfying the above property commutes. Some formulas for computing these probabilities for dihedral groups are obtained.

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Co-prime Probability for Nonabelian Metabelian Groups of Order Less than 24 and Their Related Graphs

Cited by 5 publications

References 6 publications

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Relative Coprime Probability and Graph for Some Nonabelian Groups of Small Order and Their Associated Graph Properties

The Order Product Prime Probability and Commutativity Degree for Some Finite Groups

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