On Suborbits and Graphs Associated with Action of Alternating Groups on Cartesian Product of Two Sets

Kadedesya, Stephen; Nyaga, Lewis; Rimberia, Jane

doi:10.9734/arjom/2023/v19i11763

ARJOM

2023

DOI: 10.9734/arjom/2023/v19i11763

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On Suborbits and Graphs Associated with Action of Alternating Groups on Cartesian Product of Two Sets

Stephen Kadedesya,

Lewis Nyaga,

Jane Rimberia

Abstract: In this paper, the suborbits and graphs associated with the action of direct product of two Alternating groups on the Cartesian product of two sets are studied. It is shown that the suborbits are self-paired and the associated graphs are undirected and regular with girth 3.

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2024

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Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

Orina,

Namu,

Muriuki

2024

AJAM

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In relation to group action, much research has focused on the properties of individual permutation groups acting on both ordered and unordered subsets of a set, particularly within the Alternating group and Cyclic group. However, the action of the direct product of Alternating group and Cyclic group on the Cartesian product of two sets remains largely unexplored, suggesting that some properties of this group action are still undiscovered. This research paper therefore, aims to determine the combinatorial properties - specifically transitivity and primitivity - as well as invariants which includes ranks and subdegrees of this group action. Lemmas, theorems and definitions were utilized to achieve the objectives of study with significant use of the Orbit-Stabilizer theorem and Cauchy-Frobeneus lemma. Therefore in this paper, the results shows that for any value of n ≥ 3, the group action is transitive and imprimitive. Additionally, we found out that when n = 3, the rank is 9 and the corresponding subdegrees are ones repeated nine times that is, 1, 1, 1, 1, 1, 1, 1, 1, 1. Also, for any value of n > 4, the rank is 2n with corresponding subdegrees comprising of n suborbits of size 1 and n suborbits of size (n − 1).

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Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

Orina,

Namu,

Muriuki

2024

AJAM

View full text Add to dashboard Cite

show abstract

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On Suborbits and Graphs Associated with Action of Alternating Groups on Cartesian Product of Two Sets

Abstract: In this paper, the suborbits and graphs associated with the action of direct product of two Alternating groups on the Cartesian product of two sets are studied. It is shown that the suborbits are self-paired and the associated graphs are undirected and regular with girth 3.

Cited by 1 publication

References 11 publications

Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets

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