On nonsolvable groups whose prime degree graphs have four vertices and one triangle

Document Type : Ischia Group Theory 2016

Author

Department of‎ ‎Mathematics‎, ‎Gebze Technical University‎, ‎P.O.Box 41400, Gebze‎, ‎Turkey

Abstract

‎Let $G$ be a finite group‎. ‎The prime degree graph of $G$‎, ‎denoted‎ ‎by $\Delta(G)$‎, ‎is an undirected graph whose vertex set is $\rho(G)$ and there is an edge‎ ‎between two distinct primes $p$ and $q$ if and only if $pq$ divides some irreducible‎ ‎character degree of $G$‎. ‎In general‎, ‎it seems that the prime graphs‎ ‎contain many edges and thus they should have many triangles‎, ‎so one of the cases that would be interesting is to consider those finite groups whose prime degree graphs have a small number of triangles‎. ‎In this paper we consider the case where for a nonsolvable group $G$‎, ‎$\Delta(G)$ is a connected graph which has only one triangle and four vertices‎.

Keywords

Main Subjects

References

[1] B. Huppert and W. Lempken, Simple groups of order divisible by at most four primes, Proc. of the F. Scorina Gomel State Univ., 16 (2000) 64–75.
[2] M. L. Lewis, An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math., 38 (2008) 175–211.
[3] M. L. Lewis and D. L. White, Four-vertex degree graphs of nonsolvable groups, J. Algebra, 378 (2013) 1–11.
[4] H. P. Tong-Viet, Groups whose prime graphs have no triangles, Journal of Algebra, 378 (2013) 196–206.
[5] H. P. Tong-Viet, Prime graphs of finite groups with a small number of triangles, Monatsh. Math., 175 (2014) 457–484.
[6] D. L. White, Charcater degrees of extensions of PSL(2,q) and SL(2,q), J. Group Theory, 16 (2013) 1–33.
International Journal of Group Theory
Volume 7, Issue 3 - Serial Number 3
Proceedings of the Ischia Group Theory 2016-Part III
September 2018
Pages 1-6

Files

Share

How to cite

Statistics