http://hdl.handle.net/10386/67 Tue, 07 Apr 2026 07:16:05 GMT 2026-04-07T07:16:05Z http://hdl.handle.net/10386/5383 Construction of designs and codes invariant under the group PSp4 (q), where q is a prime power Mokalapa, Clarence Kareana In this thesis, we aim to construct some designs and their codes from the projective symplectic group PSp4(q), where “q” is a prime power. We study the primitive permutation representations and conjugacy classes of projective symplectic groups and construct binary and ternary codes invariant under the group PSp4(q). To achieve this, we examine the structures of each maximal subgroup and study the conjugacy class table and character table within the group PSp4(q). We identify fixed points of the primitive actions of PSp4(q) and compute the designs parameters. We then construct codes that are invariant under the action of PSp4(q). Our investigation focuses on specific classes of maximal subgroups of PSp4(q), where we determine the design parameters using two methods known as Key-Moori Method 1 and 2. Finally, we construct codes based on these design parameters. The goal is to find the suborbits corresponding to specific maximal subgroups. To do this, we first identify some maximal subgroups of the group, then examine how the group action partitions the set into suborbits under the action of each maximal subgroup. These suborbits can help us understand the structure of the group and its actions more effectively, as outlined in Method 1. We examine the character table of the group PSp4(q), which provides information about the irreducible representations and their degrees. Using this information, we then find the corresponding permutation characters, which describe how the group acts on sets, such as conjugacy classes or coset spaces, under Method 2. By employing the techniques of examining suborbits and character table of the group PSp4(q), we are able to derive block-primitive and point-transitive 1−(v, k, λ) designs from the conjugacy classes and maximal subgroups of the group PSp4(q). We analyze 1 − (v, k, λ) design properties and construct codes that are defined by the action of groups matrices of the designs. We also determine the corresponding linear codes associated with these designs in the special case where q = 3. We examine various properties of these codes, such as their dual structure, weight distribution, and error-detection and error-correction capabilities. The codes, which are constructed using the permutation group, can be viewed as submodules of the permutation module corresponding to the action of PSp4(q). By analyzing these codes, we gain insights into their invariant properties and symmetries in relation to the group action. To demonstrate this, we focus on the finite simple group G = PSp4(3) and explore the linear and ternary codes derived from its 2- and 3-dimensional representations. Additionally, we establish connections between the combinatorial designs and the codes that remain invariant under the action of PSp4(3). Thesis (Ph.D. (Mathematics)) -- University of Limpopo, 2025 Wed, 01 Jan 2025 00:00:00 GMT http://hdl.handle.net/10386/5383 2025-01-01T00:00:00Z http://hdl.handle.net/10386/4868 On strongly paracompact and uniformly paracompact locales Gwizo, Gorden This study investigates the properties of strongly paracompact and uniformly paracompact spaces into locales. The research builds on the work of Rice [31], Frolik [15], and Borubaev [3], who expanded the concept of paracompact spaces, making it applicable to a wider range of topological situations and allowing for the use of more ideas. The main interest lies in exploring the strongly uniformly paracompact property. The dissertation looks into how recently introduced paracompactness concepts such as R paracompactness Rice [31], can be adapted into locales. In the context of a uniform space (X, U), the study defines a space as uniformly R-paracompact if every open covering has an open, uniformly locally finite refinement.. The research aims to provide a thorough exploration of these concepts, contributing to a deeper understanding of their implications within the context of uniform spaces. Thesis (M.Sc. (Mathematics)) -- University of Limpopo, 2024 Mon, 01 Jan 2024 00:00:00 GMT http://hdl.handle.net/10386/4868 2024-01-01T00:00:00Z http://hdl.handle.net/10386/4648 On P- frames and their generalisations Ngoako, Thabo David In this dissertation, we study P -frames and their generalisations. On the generalisations of P - frames we consider, in particular essential P -frames, CP -frames, almost P -frames, F -frames, F ′-frames and PF -frames. We show that a frame L is a P -frame if and only if every ideal of RL is a z-ideal. We also consider R-modules and then show that a frame L is a P -frame if and only if every RL-module is flat. Furthermore, we consider the Artin-Rees property and show that a frame L is a P -frame if and only if RL is an Artin-Rees ring. Concerning CP -frames we show, analogously to P -frames, that a frame L is a CP -frame if and only if every ideal of RcL is a zc-ideal. It turns out that in CP -frame radical ideals are precisely zc-ideals. We show, regarding F -frames, that L is an F -frame if and only if RL is a B´ezout ring. We show that L is an F -frame if and only if every ideal of RL is convex. Finally, we introduce PF -frames and show that L is a PF -frame if and only if it is an essential P -frame which is also an F -frame. Thesis (M.Sc. (Mathematics)) -- University of Limpopo, 2024 Mon, 01 Jan 2024 00:00:00 GMT http://hdl.handle.net/10386/4648 2024-01-01T00:00:00Z http://hdl.handle.net/10386/4437 Fischer-Clifford matrices and character tables of some group extensions Raboshakga, M. M. A useful way for studying the properties of a group G is to express the elements of G in terms of matrices or permutations [27]. In 1878 Cayley showed that every group G is isomorphic to a subgroup of the symmetric group Sa, where Sa is the group of all permutations on G. A representation of a group G is a homomorphism T: G-+ GL(n, F) from G into the group GL(n, F) of n x n invertible matrices over a field F. The character of G afforded by a representation T is the trace of the matrices T(g) for each g E G. The table of characters of G is called the character table of G [20]. Since the completion of the classification of finite simple groups in 1981 [4], current research work in group theory involves the study of the structures of simple groups. The structures and character tables of maximal subgroups of simple groups give substantive information about these groups. Most of the maximal subgroups of simple groups [8] and some of their constituent groups are of extension type (i.e. a group G = N.G such that N <I G and G/N 􀀃 G). In our research we are particularly interested in faithful permutation representa­tions of sporadic simple groups and their automorphism groups [27]. The Mathieu groups are examples of sporadic simple groups [32]. A permutation group G on X is said to be k-transitive on X if for any two k-tuples (x1, x2, ... , xk) and (y1, Y2, ... , Yk) of k distinct elements of X, there exists g E G such that xf = Yi, 1 :S i :S k. Apart from the symmetric groups Sn and Alternating groups An, the Mathieu groups are the only non-trivial faithful k-transitive permutation groups for k = 4, 5 [32]. Now as can be seen from the Atlas of Finite Groups [8], the Mathieu group M22 has a maximal subgroup of form 24 : S5. Likewise the Mathieu group M23 has a maximal subgroup of form 24 :A7. As part of this project we will study the groups of forms 24 : Ss and 24 : A7. We note that some groups of form pn -l : Sn, where p is prime, have been studied in [35], however the group 24 : Ss studied here is not one of those groups. This will give some information about the structures of these groups. Let m, n E N, the set of positive integers and Zm = {O, 1, ... , m - 1} be the set of residues modulo m, also considered as a cyclic group Cm of order m. One of the groups to be studied in this work is a subgroup of the symmetric group Bmn of degree m x n. For example, we know from [28] that the simplectic group SP(6, 2), which is a maximal subgroup of the Fischer group Fi22 , has a subgroup of form 25 : 85 . Further the group 25 : S5 has a subgroup isomorphic to the split extension (S3)2 : C2 or the wreath product of the symmetric group S3 of degree 3 with the cyclic group C2 of order 2. In this project, we will also study the groups (Sn)m : Cm, where mis prime and n is a positive integer. The group (Sn)m : Cm is a subgroup of the group Smn of degree m x n. Several methods for constructing the character tables of group extensions exist. However, Fischer [12] has given an effective method for constructing the character tables of some group extensions including the groups cited above. This method known as the technique of the Fischer-Clifford matrices makes use of Clifford Theory [7, 20]. Given a group extension G = H.G such that every irreducible character of H can be extended to its inertia group, for each conjugacy class of G we construct a matrix called a Fischer-Clifford matrix of G. By using the Fischer-Clifford matrices of G together with the fusion maps and the character tables of inertia factor groups of G, we are able to construct the character table of G. We note that Fischer-Clifford matrices satisfy certain properties which may be used to construct them. The method of the Fischer-Clifford matrices has been used in many works both on split and non-split extensions [1, 10, 12, 25, 28, 30, 34, 35]. Here we will use the method of Fischer-Clifford matrices to construct the character tables of the groups 24 : S5, 24 : A1, (S3)2 : C2, (S3)3 : C3, (S4)2 : C2 and ( Thesis (M.Sc. (Mathematics)) -- University of Limpopo, 2008 Tue, 01 Jan 2008 00:00:00 GMT http://hdl.handle.net/10386/4437 2008-01-01T00:00:00Z