# Recognition of the simple groups $PSL_2(q)$ by character degree graph and order

Document Type : Research Paper

Authors

1 Faculty of Mathematics and Computer science, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

2 Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 15914 Tehran, Iran

Abstract

‎‎Let $G$ be a finite group‎, ‎and $Irr(G)$ be the set of complex irreducible characters of $G$‎. ‎Let $\rho(G)$ be the set of prime divisors of character degrees of $G$‎. ‎The character degree graph of $G$‎, ‎which is denoted by $\Delta(G)$‎, ‎is a simple graph with vertex set $\rho(G)$‎, ‎and we join two vertices $r$ and $s$ by an edge if there exists a character degree of $G$ divisible by $rs$‎. ‎In this paper‎, ‎we prove that if $G$ is a finite group such that $\Delta(G)=\Delta(PSL_2(q))$ and $|G|=|PSL_2(q)|$‎, ‎then $G\cong PSL_2(q)$‎.

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#### References

[1] B. Hupp ert, Endliche Grupp en I, Die Grund lehren der Mathematischen Wissenschaften , Springer-Verlag, Berlin-New
York, 1967.
[2] B. Hupp ert, Character Theory of Finite Groups , De Gruyter Exp ositions in Mathematics, 25 , Walter de Gruyter Co.,
Berlin, (1998).
[3] I. M. Isaacs, Character theory of nite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace
Jovanovich, Publishers], New York-London, 1976.
[4] B. Khosravi, B. Khosravi and B. Khosravi, Recognition of PSL(2 ; p ) by order and some information on its character
degrees where p is a prime, Monatsh. Math. , 175 (2014) 277{282.
[5] B. Khosravi, B. Khosravi, B. Khosravi and Z. Momen, Recognition by character degree graph and order of the simple
groups of order less than 6000, Miskolc Math. Notes , 15 (2014) 537{544.
[6] B. Khosravi, B. Khosravi, B. Khosravi and Z. Momen, Recognition of the simple group PSL(2 ; p 2 ) by character degree
graph and order, Monatsh. Math. , 178 (2015) 251{257.
[7] M. L. Lewis, An overview of graphs asso ciated with character degrees and conjugacy class sizes in nite groups, Rocky
Mt. J. Math. , 38 (2008) 175{211.
[8] M. L. Lewis and J. K. McVey, Character degree graphs of automorphism groups of characteristically simple groups, J.
Group Theory, 12 (2009) 387{391.
[9] O. Manz, R. Staszewski and W. Willems, On the numb er of comp onents of a graph related to character degrees, Proc.
Amer. Math. Soc. , 103 (1988) 31{37 .
[10] A. Moreto and P. H. Tiep, Prime divisors of character degrees, J. Group Theory , 11 341-356 (2008).
[11] The GAP Group, GAP - Groups, Algorithms and Programming, Vers. 4.4.12 (2008), http://www.gap-system.org .
[12] H. P. Tong-Viet, Groups whose prime graphs have no triangles, J. Algebra , 378 (2013) 196{206.
[13] D. L. White, Character degrees of extensions of PSL 2 ( q ) and SL 2 ( q ), J. Group Theory , 16 (2013) 1{33.
[14] D. L. White, Degree graphs of simple groups, Rocky Mountain J. Math. , 39 (2009) 1713{1739.
[15] K. Zsigmondy, Zur theorie der p otenzreste, Monatsh. Math. Phys. , 3 (1892) 265{284.