http://hdl.handle.net/123456789/17457 2026-04-05T21:57:13Z http://hdl.handle.net/123456789/17467 ON SKEW QUASI-P-CLASS (Q) OPERATORS, POSIMETRICALLY EQUIVALENT OPERATORS, AND MUTUALLY CLASS (Q) OPERATORS WANJALA VICTOR WAFULA The study of class (Q) operators on Hilbert spaces has been exploited into various classes such as Quasi class (Q) , M-Quasi class (Q) , (n+k)-Class (Q) , Almost class (Q) and (α, β)- class (Q) among others . Results have been proved showing that some of these classes converge to the strong operator topology and results striking relationships between these classes and other general classes were achieved . However , little has been done to expand the results of class (Q) operators into the class of skew-Quasi-p-class (Q) . Hence, in this study, we introduce the category of Skew Quasi-p-class (Q) operators. We examine the fundamental characteristics of this class and establish its connection with other classes, such as quasi-p-normal operators . We also introduce the class of Posimetrically equivalent operators which is a generalization of Metrically equivalent operators , we characterize this class in terms of Complex symmetric operators and study their relations with other equivalence relations such as the class of n Metrically equivalent operators . We finally introduce the class of Mutually class (Q) operators . Furthermore, we explore the interrelation between this class and other classes in a comprehensive manner . The methodology used include but not limited to , properties of operators like unitary operators , quasi-p-normal operators and skew-adjoint operators . Results shows that the class of skew quasi-p-class (Q) operators have Bishop’s property and that they are isoloid and polaroid ; Posimetrically equivalent operators are closed under scalar multiplication and Mutually class (Q) operators are related to class (Q) operators . The study of these classes of (Q) operators will be helpful in the telecommunication industry by generalizing the allocation of network resources basing on priority of network . As a result , high-priority traffic such as video and voice will be given more bandwidth by being transmitted with lower packet loss and delay . 2024-01-01T00:00:00Z http://hdl.handle.net/123456789/17458 CLASSIFICATION OF INTERNAL STRUCTURES OF SOME GROUPS OF EXTENSION USING MODULAR REPRESENTATION METHOD JANET LILIAN MAINA In the communication process, a sender encodes a message which is then send through a communication channel. There could be a barrier in the channel such that the mes sage gets distorted before it reaches the recipient. A solution is in the need for con struction of more optimal codes for error detecting and correcting. This research fo cused on representing the internal structures of groups of extensions using modular rep resentation method. Specifically, it examined the maximal subgroups of four groups: 0 + 8 (2) : 2, L3(4) : 2, L3(4) : 22 and L3(3) : 2. For each of these groups, detailed anal ysis was provided on the irreducible representations of their maximal subgroups, across varying representation degrees. The key goal was to classify internal structures of the groups using modular representations method. The specific objectives were to classify maximal subgroups of the groups of extension , enumerate linear codes from the maximal subgroups, construct lattice diagrams of linear codes obtained and analyze the proper ties of linear codes and designs constructed using the modular representation method. By decomposing into irreducible constituents, the work uncovered new linkages between representation theory, finite group extensions, and combinatorial designs. For the group 0 + 8 (2) : 2, representations of degree 120, 135, and 960 across multiple maximal subgroups were explored. Similarly, representations ranging from degree 21 to 336 were analyzed for the maximal subgroups under L3(4) : 2 and L3(4) : 22 . Finally, representations up to degree 234 were examined among the maximal subgroups under L3(3) : 2. In mapping these finite groups through their maximal subgroups representations systematically, the work contributes enhanced understanding of how extended finite groups can be classified internally based on modular representation structures. Findings fill a gap in current group representation theory literature related to certain orders of linear groups of extensions. Outcomes point to opportunities for further exploration into additional families of finite groups using similar representation mapping techniques. Findings from the research on this classification of linear codes and designs for error correction gets their applicability in digital communication, data storage and cryptography. 2024-01-01T00:00:00Z