https://repository.cuk.ac.ke/handle/123456789/625 2026-04-20T21:59:30Z https://repository.cuk.ac.ke/handle/123456789/1898 Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets Gikunju, David Muriuki; Nyaga, Lewis Namu The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2. Studies suborbital graphs under group actions 2024-10-18T00:00:00Z https://repository.cuk.ac.ke/handle/123456789/1897 Combinatorial Properties and Invariants Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets Mokaya John, Victor; Nyaga, Lewis Namu; Gikunju, David Muriuki Group theory, an area in mathematics, has undergone extensive research, but not exhaustively. In group theory, especially symmetric groups have been studied in terms of properties and actions. In this research, two groups of the Symmetric Group (Alternating and Dihedral Groups) are studied in terms of their properties and also their direct product between them. Also, ordered sets are studied in this research where their Cartesian product is looked at. Finally, this research combines the direct product of two symmetric groups (alternating group and dihedral group) and the Cartesian product of two ordered sets (X×Y) by group action. This research therefore focuses on determining the combinatorial properties (transitivity and primitivity), invariants (ranks and subdegrees), and structures (suborbital graphs) of this group action. To accomplish this, orbit-stabilizer theorem is used to compute transitivity, block action concept is applied to determine primitivity, and Cauchy-Frobenius lemma is used to compute ranks and subdegrees. For n ≥ 3, it has been confirmed that the group action is transitive and also imprimitive. The rank of this group action is 6 and the subdegrees are obtained using the formula of Theorem 3.3. This research adds new concepts in group theory which will be useful in other areas like graph theory and also application in real life situations. Group action structures and invariants (alternating × dihedral) 2024-12-18T00:00:00Z https://repository.cuk.ac.ke/handle/123456789/1896 Structures Associated with the Direct Product of Alternating and Dihedral Groups Acting on the Cartesian Product of Two Sets Victor, John Mokaya; Namu, Nyaga Lewis; Muriuki, Gikunju David Whenever a permutation group acts on a set, combinatorial and invariant properties, and mathematical structures that result from this group action are studied. Various mathematicians have studied these properties over time using different groups acting on both ordered and unordered sets. The combinatorial properties (transitivity and primitivity) and invariants (ranks and subdegrees) of the direct product between alternating and dihedral groups acting on the Cartesian product of two sets have already been studied and it was found out that the group action is transitive, imprimitive, therankis6, and sub degrees are obtained according to theorem 2.3. This research seeks to extend this by constructing and analyzing the properties (simple/multigraph, self-pairedness, connectedness, degree of the vertex, girth, and directedness) of these mathematical structures (suborbital graphs) that result from the group action. This research for n ≥ 3, suborbital graphs can be classified into three categories; First, those constructed when only the first components of the vertex set are identical and second, those when only the second components of the vertex set are identical. The suborbital graphs of the first and second category are simple, self-paired, have n− disconnected components, are regular with degree n − 1 and girth is 3. The third category of suborbital graphs in which neither the first nor the second components of the vertex set are identical and they are; simple, self-paired, connected, regular with degree of vertex varying from graph to graph, and girth 3. Structural results for alternating × dihedral direct product group actions 2024-12-01T00:00:00Z https://repository.cuk.ac.ke/handle/123456789/1895 Revive the interest to learn math: A psychological approach. Mukudi, Fidelis Musena A short book on learning math using psychological approaches 2024-11-20T00:00:00Z