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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations
- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2022
- ISSN
- 0126-6705
- eISSN
- 2180-4206
- DOI
- 10.1007/s40840-022-01329-6
- Publisher site
- See Article on Publisher Site
Abstract
Let G be a finite group. For an irreducible character χ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi $$\end{document} of G, the number χc(1)=[G:kerχ]χ(1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi ^c(1)=\frac{[G:\mathrm{ker}\chi ]}{\chi (1)}$$\end{document} is called the co-degree of χ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\chi $$\end{document}. Let Codeg(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm{Codeg}(G)$$\end{document} denote the set of co-degrees of the irreducible (complex) characters of G. The co-degree graph of G is defined as the simple undirected graph whose vertices are prime divisors of the numbers in Codeg(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm{Codeg}(G)$$\end{document} and two distinct vertices p and q are joined by an edge if and only if pq divides some number in Codeg(G)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm{Codeg}(G)$$\end{document}. In this paper, we first study the finite non-solvable groups whose co-degree graphs have no complete vertices. Then, we show that if the co-degree graph of G has no complete vertices, then for every x∈G\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x \in G$$\end{document}, G admits an irreducible character whose co-degree is divisible by o(x), the order of x. Finally, we inspect the finite groups whose co-degree graphs are m-regular.
Journal
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Sep 1, 2022
Keywords: The co-degree of a character; Co-degree graph; Order element; Complete vertex; 20C15; 20D10; 20D60