Classification of Internal Structures of Some Groups of Extension Using Modular Representation Method

• Author(s): Janet Lilian Maina
• Paper ID: 1707090
• Page: 17-23
• Published Date: 03-02-2025
• Published In: Iconic Research And Engineering Journals
• Publisher: IRE Journals
• e-ISSN: 2456-8880
• Volume/Issue: Volume 8 Issue 8 February-2025

Abstract

This research investigates the internal structures of specific group extensions using modular representation theory, with applications to error detection and correction in digital communications. The study focuses on four significant group extensions: O??(2):2, L?(4):2, L?(4):2², and L?(3):2, analyzing their maximal subgroups through different representation degrees to understand their structural properties and potential applications in coding theory. Through systematic computational analysis using the GAP system and methodologies derived from classical representation theory, we identified distinct patterns in the representation decomposition of these groups in characteristic 2. Our findings revealed that O??(2):2 possesses 24 conjugacy classes and 16 maximal subgroups, with representation patterns that directly influence code construction efficiency. The study established new relationships between group structure and code parameters, leading to the development of error-correcting codes with superior performance metrics compared to traditional approaches. The constructed codes demonstrated significant improvements in error correction capabilities, achieving rates of up to 97.2% while maintaining computational efficiency. Implementation analysis showed promising results for practical applications, with average encoding times of 0.45ms and modest memory requirements. These findings contribute to both theoretical understanding of group extensions and practical advancements in coding theory, offering new perspectives on the relationship between abstract algebraic structures and digital communication systems. This research advances the field by bridging pure mathematical theory with practical applications, providing a foundation for future developments in both group theory and coding theory. The results suggest promising directions for generalizing these methods to broader classes of group extensions and optimizing code construction for modern communication systems.

Keywords

group extensions, modular representation theory, error-correcting codes, finite groups, digital communications

Citations

IRE Journals:
Janet Lilian Maina "Classification of Internal Structures of Some Groups of Extension Using Modular Representation Method" Iconic Research And Engineering Journals Volume 8 Issue 8 2025 Page 17-23

IEEE:
Janet Lilian Maina "Classification of Internal Structures of Some Groups of Extension Using Modular Representation Method" Iconic Research And Engineering Journals, 8(8)