Bounding the number of vertices in the degree graph of a finite group
| Autor: | Lucia Sanus, Emanuele Pacifici, Zeinab Akhlaghi, Silvio Dolfi |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Finite group
Algebra and Number Theory 20C15 010102 general mathematics Group Theory (math.GR) 01 natural sciences Upper and lower bounds Graph Vertex (geometry) Combinatorics Bounding overwatch 0103 physical sciences FOS: Mathematics Maximum size 010307 mathematical physics 0101 mathematics Undirected graph Mathematics - Group Theory Clique number Mathematics |
| Zdroj: | Journal of Pure and Applied Algebra. 224:725-731 |
| ISSN: | 0022-4049 |
| DOI: | 10.1016/j.jpaa.2019.06.006 |
| Popis: | Let G be a finite group, and let cd ( G ) denote the set of degrees of the irreducible complex characters of G . The degree graph Δ ( G ) of G is defined as the simple undirected graph whose vertex set V ( G ) consists of the prime divisors of the numbers in cd ( G ) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd ( G ) . In this note, we provide an upper bound on the size of V ( G ) in terms of the clique number ω ( G ) (i.e., the maximum size of a subset of V ( G ) inducing a complete subgraph) of Δ ( G ) . Namely, we show that | V ( G ) | ≤ max { 2 ω ( G ) + 1 , 3 ω ( G ) − 4 } . Examples are given in order to show that the bound is best possible. This completes the analysis carried out in [1] where the solvable case was treated, extends the results in [3] , [4] , [9] , and answers a question posed by the first author and H.P. Tong-Viet in [4] . |
| Databáze: | OpenAIRE |
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