Enhanced Algorithm for Modular Isomorphism Problem Resolution in Small Group Orders

Abstract

In this paper, we propose an enhanced algorithmic approach for resolving the Modular Isomorphism Problem (MIP) for groups of small orders. Building upon Eick's algorithm, our improvement obviates the need for computing the full augmentation ideal, thereby significantly enhancing computational efficiency. Through our computations, we provide affirmative resolutions to the MIP for groups of order 37 and substantially reduce the computational burden for groups of order 56. Furthermore, we present a comprehensive analysis of the recent counterexamples to the MIP discovered by García-Lucas, Margolis, and del Río, demonstrating that these counterexamples represent the sole instances of 2- or 3-generated counterexamples of order 29. Additionally, we offer a rigorous proof for an observation by Bagiński, which aids in the elimination of computationally challenging cases. Our research not only advances the theoretical understanding of the MIP but also provides practical tools for its resolution in small group orders.In this article, as a network manager, we're concerned with setting up means of access control, and to do this, we have to square a kind of circle : simplicity for the user, reliability of the mechanisms, high level of security, all while using available standards as much as possible.