Ranks, Subdegrees, Suborbital Graphs and Cycle Indices Associated with the Product Action of Affine Groups

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dc.contributor.advisorKamuti Irerien_US
dc.contributor.advisorPatrick Kimanien_US
dc.contributor.authorAgwanda, Siahi Maxwell
dc.date.accessioned2022-03-30T11:33:06Z
dc.date.available2022-03-30T11:33:06Z
dc.date.issued2021
dc.descriptionA Project submitted in Partial Fullfilment of the Requirements for Award of Masters Degree (Pure Mathematics) in the School of Pure and Applied Sciences of Kenyatta University, October 2021en_US
dc.description.abstractMany scholars have studied the ranks, subdegrees, cycle index and graphs of the action of the groups Cn,Dn and Aff(q) on a set X, where X = {1, 2, . . . , n} leaving out product actions. Recently, Kangogo (2015) studied the action of affine group over Galois field. The action of Aff(q1) × Aff(q2) on GF(q1) × GF(q2) and Aff(q1) × Aff(q2) × Aff(q3) on GF(q1) × GF(q2) × GF(q3) has not been studied. Using the definition of product action of orbits, the properties of the action of Aff(q1) × Aff(q2) on GF(q1)×GF(q2) and Aff(q1)×Aff(q2)×Aff(q3) on GF(q1)×GF(q2)× GF(q3) were studied and the rank was found to be 2k, where k = 2, 3 is the number of affine groups in the cross product. The subdegrees were found to be 1, (q1−1), (q2− 1), (q1 − 1)(q2 − 1) and 1, (q1 − 1), (q2 − 1), (q3 − 1), (q1 − 1)(q2 − 1), (q1 − 1)(q3 − 1), (q2−1)(q3−1), (q1−1)(q2−1)(q3−1) respectively. The corresponding non trivial graphs of Aff(q1)×Aff(q2) on GF(q1)×GF(q2) and Aff(q1)×Aff(q2)×Aff(q3) on GF(q1) × GF(q2) × GF(q3) were constructed using Sim’s procedure and were found to have a girth of 0, 3, 6 and 0, 3, 4, 6 respectively. Finally, cycle index were determined by first determining the cycle index of Aff(q) acting on GF(q) and then using multiplication of monomials to get the cycle index of the product action. The cycle indices have applications in chemistry when counting isomers.The graphs constructed provide useful information to graph theorist. Connectivity in graphs helps biologists to explain how the different parts of the brain are connected. The results have been represented in form of theorems and graphs.en_US
dc.description.sponsorshipKenyatta Universityen_US
dc.identifier.urihttp://ir-library.ku.ac.ke/handle/123456789/23431
dc.language.isoenen_US
dc.publisherKenyatta Universityen_US
dc.subjectRanksen_US
dc.subjectSubdegrees,en_US
dc.subjectSuborbital Graphsen_US
dc.subjectCycle Indices Associateden_US
dc.subjectProduct Actionen_US
dc.subjectAffine Groupsen_US
dc.titleRanks, Subdegrees, Suborbital Graphs and Cycle Indices Associated with the Product Action of Affine Groupsen_US
dc.typeThesisen_US
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