Finite Groups Whose Character Degree Graphs Coincide With Their Prime Graphs

• Temha Erkoç
• Utku Yilmaztürk
• İsmail Ş. Güloğlu
Article

Abstract

In the literature, there are several graphs related to a finite group G. Two of them are the character degree graph, denoted by Δ(G), and the prime graph, Γ(G). In this paper we classify all finite groups whose character degree graphs are disconnected and coincide with their prime graphs. As a corollary, we find all finite groups whose character degree graphs are square and coincide with their prime graphs.

Keywords

finite groups character degree graph prime graph

20C15

Preview

Unable to display preview. Download preview PDF.

References

1. [1]
B. Huppert: Endliche Gruppen I. Die Grundlehren der Mathematischen Wissenschaften 134, Springer, Berlin, 1967. (In German.)Google Scholar
2. [2]
M. L. Lewis: Solvable groups whose degree graphs have two connected components. J. Group Theory 4 (2001), 255–275.
3. [3]
M. L. Lewis, Q. Meng: Square character degree graphs yield direct products. J. Algebra 349 (2012), 185–200.
4. [4]
M. L. Lewis, D. L. White: Connectedness of degree graphs of nonsolvable groups. J. Algebra 266 (2003), 51–76 (2003); corrigendum ibid. 290 (2005), 594–598.
5. [5]
C. P. Morresi Zuccari: Character degree graphs with no complete vertices. J. Algebra 353 (2012), 22–30.
6. [6]
D. L. White: Degree graphs of simple linear and unitary groups. Commun. Algebra 34 (2006), 2907–2921.
7. [7]
J. S. Williams: Prime graph components of finite groups. J. Algebra 69 (1981), 487–513.

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

• Temha Erkoç
• 1
• Utku Yilmaztürk
• 1
• İsmail Ş. Güloğlu
• 2
1. 1.Department of Mathematics Faculty of ScienceIstanbul UniversityFatih, IstanbulTurkey
2. 2.Department of MathematicsDoğuş UniversityAcıbadem, IstanbulTurkey