CLASSIFICATION OF INTERNAL STRUCTURES OF SOME GROUPS OF EXTENSION USING MODULAR REPRESENTATION METHOD

Abstract

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.