Parallelisms of PG(3,4) invariant under an elementary abelian group of order 4

IF 0.6 4区工程技术 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Applicable Algebra in Engineering Communication and Computing Pub Date : 2022-06-22 DOI:10.1007/s00200-022-00562-7

Anton Betten, Svetlana Topalova, Stela Zhelezova

引用次数: 1

Abstract

This paper is a contribution to the classification of parallelisms in three-dimensional projective spaces over small finite fields of order q by computer. The smallest space in which parallelisms have not yet been classified is for \(q=4.\) Partial results are available. The parallelisms admitting a nontrivial automorphism of odd prime order are known. Moreover, much is known about the case of parallelisms of \({{\mathrm{PG}}}(3,4)\) whose automorphism group is a two group. Namely, everything is known for two of the three possible groups of order two, as well as for cyclic groups of order 4. The present paper will settle the case of parallelisms whose automorphism group is elementary abelian of order 4. This leaves open the cases of parallelisms whose full automorphism groups are either trivial or a specific group of order two.

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4阶初等阿贝尔群下PG(3,4)不变量的并行性

本文对用计算机对q阶小有限域上三维射影空间中的平行性进行分类作出了贡献。平行性尚未分类的最小空间是for（q=4.\）。部分结果可用。承认奇素数阶非平凡自同构的平行性是已知的。此外，关于自同构群是二群的\（｛｛\mathrm｛PG｝｝｝｝（3，4）\）的平行性的情况已经知道很多。也就是说，对于三个可能的二阶群中的两个，以及对于四阶循环群，一切都是已知的。本文将解决自同构群为4阶初等阿贝尔的平行同构的情况。这使得完全自同构群要么是平凡的，要么是二阶的特定群的平行主义的情况是开放的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Applicable Algebra in Engineering Communication and Computing 工程技术-计算机：跨学科应用

CiteScore

2.90

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Algebra is a common language for many scientific domains. In developing this language mathematicians prove theorems and design methods which demonstrate the applicability of algebra. Using this language scientists in many fields find algebra indispensable to create methods, techniques and tools to solve their specific problems. Applicable Algebra in Engineering, Communication and Computing will publish mathematically rigorous, original research papers reporting on algebraic methods and techniques relevant to all domains concerned with computers, intelligent systems and communications. Its scope includes, but is not limited to, vision, robotics, system design, fault tolerance and dependability of systems, VLSI technology, signal processing, signal theory, coding, error control techniques, cryptography, protocol specification, networks, software engineering, arithmetics, algorithms, complexity, computer algebra, programming languages, logic and functional programming, algebraic specification, term rewriting systems, theorem proving, graphics, modeling, knowledge engineering, expert systems, and artificial intelligence methodology. Purely theoretical papers will not primarily be sought, but papers dealing with problems in such domains as commutative or non-commutative algebra, group theory, field theory, or real algebraic geometry, which are of interest for applications in the above mentioned fields are relevant for this journal. On the practical side, technology and know-how transfer papers from engineering which either stimulate or illustrate research in applicable algebra are within the scope of the journal.