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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.
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References
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Bagiński, A. (2018). Observations on the modular isomorphism problem. arXiv preprint arXiv:1811.02538.
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Udoaka O.G., Omelebele j. and Udo-akpan I. U., (2022). Rank of identity Difference Transformation Semigroup., Int. journal of pure mathematics, vol. 9,
Frank E. A. and Udoaka O. G., Finite Semi-group Modulo and Its Application to Symmetric Cryptography. INTERNATIONAL JOURNAL OF PURE MATHEMATICS
DOI: 10.46300/91019.2022.9.13.
Udoaka, O. G., (2022) Generators and inner automorphism.. THE COLLOQUIUM -A Multi-disciplinary Thematc Policy Journal www.ccsonlinejournals.com Volume 10 , Number 1, 2022 Pages 102 -111 CC-BY-NC-SA 4.0 International Print ISSN : 2971-6624 eISSN: 2971-6632.
Udoaka O. G, Tom O. and Musa A., (2023). On Idempotent Elements in Quasi-Idempotent Generated Semigroup. 2023 IJRTI | Volume 8, Issue 11 | ISSN: 2456-3315, international Journal for Research Trends and Innovation (www.ijrti.org)
Udoaka O. G.,(2023). Rank of some Semigroups. International Journal of Applied Science and Mathematical Theory E- ISSN 2489-009X P-ISSN 2695-1908, Vol. 9 No.3. www.iiardjournals.org
Udoaka Otobong G. and Udoakpan I. U. (2024) "Exploration of Symmetric Groups: Cayley Tables, Subgroup Analysis, and Real-World Applications in Card Tricks Scholars Journal of Physics, Mathematics and Statistics Abbreviated key title: Sch J Phys Math Stat. ISSN 2393-8064 (Online) |ISSN 2393-8056 (Print) Publisher: SAS Publishers
Ndubuisi R. U., Shum K. P., Udoaka O. G. and Abubakar R. B.,(2019) . On Homomorphisms (Good Homomorphisms) Between Completely J^∘-Simple Semigroups, Canadian Journal of Pure and Applied Sciences. Vol. 13, No. 2, pp. 4793-4797, Online ISSN: 1920-3853; Print ISSN: 1715-9997. Available online at www.cjpas.net
Michael N. John, Edet, Effiong, & Otobong G. Udoaka. (2023). On Finding B-Algebras Generated By Modulo Integer Groups