A new characterization of simple K3-groups using same-order type

Authors

DOI:

https://doi.org/10.5902/2179460X70082

Keywords:

Element order, Same-order type, Characterization, Simple group, Kn-group simple

Abstract

Let G be a group, define an equivalence relation ∼ as below:

∀ g, h ∈ G, g ∼ h ⇐⇒|g| = |h|

the set of sizes of equivalence classes with respect to this relation is called the same-order type of G and denoted by α(G). And G is said a αn-group if |α(G)| = n. Let π(G) be the set of prime divisors of the order of G. A simple group of G is called a simple Kn-group if |π(G)| = n. We give a new characterization of simple K3-groups using same-order type. Indeed we prove that a nonabelian simple group G has same-order type {r, m, n, k, l} if and only if G ≅ PSL(2,q), with q = 7, 8 or 9. This result generalizes the main results in (4), (6) and (8). Moreover based on the main result in (8) we have the natural question: Let S be a nonabelian simple αn-group and G a αn-group such that |S| = |G|. Then S ≅ G. In this paper with a counterexample we give a negative answer to this question.

Downloads

Download data is not yet available.

Author Biographies

Igor dos Santos Lima, University of Brasília

Professor at the University of Brasília - MAT/UnB. Bachelor's degree in Mathematics from the University of Brasília - UnB. Master in Mathematics from the University of Brasília. PhD in Mathematics from the State University of Campinas. Post-Doctorate at the University of Brasília. He has experience in the areas of Mathematics Education and Mathematics, with an emphasis on Algebra. Working at the New High School Observatory, Active Methodologies, Internship Supervised and Pedagogical Residency. Working on group theory on the following topics: groups Bianchi, GAP, sophic groups, group centralizers (rings), finite group covers, maximal subgroups, separability under conjugation, Bass-Serre Theory (groups in agitation graphs), graphs defined over groups, locally finite groups, Artin groups, homology and cohomology of groups, profinite and pro-p groups and profinite and pro-p completions of groups and Poincaré duality groups.

Josyane dos Santos Pereira, University of Brasília

Graduated in Mathematics at the University of Brasília. Undertook Scientific Initiation under the guidance of Prof. Igor Lima, studying groups with a finite number of conjugation classes and element centralizers, addressing the Taghvasani-Zarrin Conjecture. Participates in the Teaching Laboratories Monitoring project under the guidance of professor Regina Pina. She was part of the pedagogical residency under the guidance of Igor Lima and Rui Seimetz. 

References

ANABANTI. C. S. A counterexample to Zarrin’s Conjecture on sizes of finite nonabelian simple groups in relation to involution sizes. Archiv der Mathematik. Disponível em: https://link.springer.com/article/10.1007/ s00013-018-1265-y, (2019) 225–226. Access in: 07 fev.2023.

GAP Group (2019). GAP – Groups, Algorithms, and Programming, Version 4.10.2. https://www.gap-system.org

HERZG M. On finite simple groups of order divisible by three primes only. Disponível em: https://www. sciencedirect.com/science/article/pii/0021869368900884?via%3DihubJ. Algebra, 10 (1968), 383–388. Access in: 07 fev. 2023.

KHATAMI M., KHOSRAVI B., AKHLAGHI Z. A new characterization for some linear groups. Disponível em: https://link.springer.com/article/10.1007/s00605-009-0168-1 Monatsh. Math., (2011), 39-50. Access in: 07 fev.2023.

MALINOWSKAI. A. Finite groups with few normalizers or involutions. Disponível em: https://link.springer. com/article/10.1007/s00013-018-1290-x Archiv der Mathematik, (2019), 459–465. Access in: 07 fev.2023.

SHEN R. On groups with given same-order type. Disponível em: https://www.researchgate.net/publication/ 254321284_On_Groups_with_Given_Same-Order_Types Comm. Algebra, (2012), 2140-2150. Access in: 07 fev.2023.

SHEN R., SHAO C., JIANG Q., MAZUROV W. A new characterization A5. Disponível em: https://www.researchgate. net/publication/225666622_A_new_characterization_of_A5 Monatsh. Math., (2010), 337-341.Access in: 07 fev.2023.

TAGHVASANI L. J. and ZARRIN M. A characterization of A5 by its same-order type. Disponível em: https://link.springer.com/article/10.1007/s00605-016-0950-9 Monatsh. Math., (2017), 731-736. Access in: 07 fev.2023.

ZARRIN M. A counterexample to Herzog’s Conjecture on the number of involutions. Disponível em: https://link.springer.com/article/10.1007/s00013-018-1195-8 Arch. Math., (2018), 349–351. Access in: 07 fev.2023.

Downloads

Published

2023-10-20

How to Cite

Lima, I. dos S., & Pereira, J. dos S. (2023). A new characterization of simple K3-groups using same-order type. Ciência E Natura, 45, e23. https://doi.org/10.5902/2179460X70082

Issue

Vol. 45 (2023): Continuous Publication

Section

Mathematics

License

Copyright (c) 2023 Ciência e Natura

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

To access the DECLARATION AND TRANSFER OF COPYRIGHT AUTHOR’S DECLARATION AND COPYRIGHT LICENSE click here.

 

Ethical Guidelines for Journal Publication

The Ciência e Natura journal is committed to ensuring ethics in publication and quality of articles. 

Conformance to standards of ethical behavior is therefore expected of all parties involved: Authors, Editors, Reviewers, and the Publisher.

In particular, 

Authors: Authors should present an objective discussion of the significance of research work as well as sufficient detail and references to permit others to replicate the experiments. Fraudulent or knowingly inaccurate statements constitute unethical behavior and are unacceptable. Review Articles should also be objective, comprehensive, and accurate accounts of the state of the art. The Authors should ensure that their work is entirely original works, and if the work and/or words of others have been used, this has been appropriately acknowledged. Plagiarism in all its forms constitutes unethical publishing behavior and is unacceptable. Submitting the same manuscript to more than one journal concurrently constitutes unethical publishing behavior and is unacceptable. Authors should not submit articles describing essentially the same research to more than one journal. The corresponding Author should ensure that there is a full consensus of all Co-authors in approving the final version of the paper and its submission for publication.

Editors: Editors should evaluate manuscripts exclusively on the basis of their academic merit. An Editor must not use unpublished information in the editor's own research without the express written consent of the Author. Editors should take reasonable responsive measures when ethical complaints have been presented concerning a submitted manuscript or published paper.

Reviewers: Any manuscripts received for review must be treated as confidential documents. Privileged information or ideas obtained through peer review must be kept confidential and not used for personal advantage. Reviewers should be conducted objectively, and observations should be formulated clearly with supporting arguments, so that Authors can use them for improving the paper. Any selected Reviewer who feels unqualified to review the research reported in a manuscript or knows that its prompt review will be impossible should notify the Editor and excuse himself from the review process. Reviewers should not consider manuscripts in which they have conflicts of interest resulting from competitive, collaborative, or other relationships or connections with any of the authors, companies, or institutions connected to the papers.