2-群的强完全映射

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2025-05-23 DOI:10.1016/j.disc.2025.114568

Reza Akhtar , Jacob Charboneau , Stephen M. Gagola III

{"title":"2-群的强完全映射","authors":"Reza Akhtar , Jacob Charboneau , Stephen M. Gagola III","doi":"10.1016/j.disc.2025.114568","DOIUrl":null,"url":null,"abstract":"<div><div>A strong complete mapping for a group <em>G</em> is a bijection <span><math><mi>φ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></math></span> such that the maps <span><math><mi>x</mi><mo>↦</mo><mi>x</mi><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>x</mi><mo>↦</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are also bijections. Groups admitting a strong complete mapping are important to the study of orthogonality problems for Latin squares and group sequencings, among other applications. In previous work we showed that a finite 3-group that contains no cyclic subgroup of index 3 is strongly admissible. In this article, we employ a substantially different strategy to show that a finite 2-group that contains no cyclic subgroup of index 4 is strongly admissible.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 10","pages":"Article 114568"},"PeriodicalIF":0.7000,"publicationDate":"2025-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Strong complete mappings for 2-groups\",\"authors\":\"Reza Akhtar , Jacob Charboneau , Stephen M. Gagola III\",\"doi\":\"10.1016/j.disc.2025.114568\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>A strong complete mapping for a group <em>G</em> is a bijection <span><math><mi>φ</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>G</mi></math></span> such that the maps <span><math><mi>x</mi><mo>↦</mo><mi>x</mi><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>x</mi><mo>↦</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>φ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are also bijections. Groups admitting a strong complete mapping are important to the study of orthogonality problems for Latin squares and group sequencings, among other applications. In previous work we showed that a finite 3-group that contains no cyclic subgroup of index 3 is strongly admissible. In this article, we employ a substantially different strategy to show that a finite 2-group that contains no cyclic subgroup of index 4 is strongly admissible.</div></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":\"348 10\",\"pages\":\"Article 114568\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2025-05-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X25001761\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X25001761","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

群G的一个强完全映射是一个双射φ:G→G，使得映射x∑xφ(x)和x∑x−1φ(x)也是双射。承认强完全映射的群对拉丁方的正交性问题和群序列的研究以及其他应用具有重要意义。在以前的工作中，我们证明了不含指标3的循环子群的有限3群是强可容许的。在本文中，我们采用了一种完全不同的策略来证明不包含索引4的循环子群的有限2群是强可容许的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Strong complete mappings for 2-groups

A strong complete mapping for a group G is a bijection

φ : G \to G

such that the maps

x \mapsto x φ (x)

and

x \mapsto x^{- 1} φ (x)

are also bijections. Groups admitting a strong complete mapping are important to the study of orthogonality problems for Latin squares and group sequencings, among other applications. In previous work we showed that a finite 3-group that contains no cyclic subgroup of index 3 is strongly admissible. In this article, we employ a substantially different strategy to show that a finite 2-group that contains no cyclic subgroup of index 4 is strongly admissible.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.