‎$‎4‎$‎-Regular prime graphs of nonsolvable groups

Document Type : Research Paper

Authors

1 School of Mathematics, Statistics and Actuarial Science, Maseno University, Kenya

2 Department of Pure and‎ ‎Mathematics‎, ‎Maseno University‎, ‎P.O‎. ‎Box 333, Maseno‎, ‎Kenya

Abstract

Let $G$ be a finite group and cd$(G)$ denote the character degree set for $G$. The prime graph $Δ(G)$ is a simple graph whose vertex set consists of prime divisors of elements in cd$(G)$, denoted $\rho(G)$. Two primes $p,q\in \rho(G)$ are adjacent in $Δ(G)$ if and only if $pq|a$ for some $a\in cd(G)$. We determine which simple $4$-regular graphs occur as prime graphs for some finite nonsolvable group.

Keywords

Main Subjects

References

[1] Z. Akhlaghi, K. Khedri and B. Taeri, Finite groups with K5-free prime graphs, Preprint.
[2] Z. Akhlaghi and H. P. Tong-Viet, Finite groups with K4-free prime graphs, Algebr. Represent. Theory, 18 (2015)
235–256.
[3] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of finite groups: maximal subgroups
and ordinary characters for simple groups, Clarendon Press, Oxford, England, 1985.
[4] M. Bianchi, D. Chillag, M. Lewis and E. Pacifici, Character degree graphs that are complete graphs, Proceedings of
the AMS, 135 (2007) 671–676.
[5] R. Guralnick, Subgroups of prime power index in a simple group, J. Algebra, 81 (1983) 304–311. http://dx.doi.
org/10.1016/0021-8693(83)90190-4.
[6] B. Huppert, Endliche gruppen I, Springer-Verlag, Berlin, 1983.
[7] 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.
[8] I. M. Isaacs, Character theory of finite groups, AMS Chelsea, 2006.
[9] G. Karpilovsky, The Schur multiplier, Clarendon Press, Oxford, 1987.
[10] M. L. Lewis and D. L. White, Connectedness of degree graphs of nonsolvable groups, J. Algebra, 266 (2003) 51–76.
[11] M. L. Lewis and D. L. White, Diameters of degree graphs of nonsolvable groups, J. Algebra, 283 (2005) 80–93.
[12] M. L. Lewis and D. L. White, Diameters of degree graphs of nonsolvable groups II, J. Algebra, 312 (2007) 634–649.
[13] M. L. Lewis and J. K. McVey, Character degree graphs of automorphism groups of characteristically simple groups,
J. Group Theory, 12 (2009) 387–391.
[14] M. L. Lewis and D. L. White, Four-vertex degree graphs of nonsolvable groups, J. Algebra, 378 (2013) 1–11.
[15] M. L. Lewis, Classifying character degree graphs with five vertices, In: Finite Groups 2003, (2004) 247–265.
[16] M. L. Lewis and Q. Meng, Square character degree graphs yield direct products, J. Algebra, 349 (2012) 185–200.
[17] K. Magaard and H. P. Tong-Viet, Character degree sums in finite nonsolvable groups, J. Group Theory, 14 (2011)
53–57.
[18] O. Manz, W. Willems and T. R. Wolf, The diameter of the character degree graph, J. Reine Angew. Math., 402
(1989) 181–198.
[19] J. K. Mc Vey, Bounding graph diameters of nonsolvable groups,J. Algebra, 282 (2004) 260–277.
[20] M. Meringer, Fast generation of regular graphs and construction of cages, J. Group Theory, 30 (1999) 137–146.
[21] A. Moret´o, Complex group algebras of finite groups: Brauer’s problem 1, Adv. Math., 208 (2007) 236–248.
[22] A. Moret´o and P. H. Tiep, Prime divisors of character degrees, J. Group Theory, 11 (2008) 341–356.
[23] P. P. P´alfy, On the character degree graph of solvable groups I: three primes, Period. Math. Hungar., 36 (1998)
61–65.
[24] P. P. P´alfy, On the character degree graph of solvable groups, II: disconnected graphs, Studia Sci. Math. Hungar.,
38 (2001) 339–355.
[25] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.6; 2016. http://www.gap-system.org.
[26] H. P. Tong-Viet, Finite groups whose prime graphs are regular, J. Algebra, 397 (2014) 18–31.
[27] H. P. Tong-Viet, Groups whose prime graphs have no triangles, J. Algebra, 378 (2013) 196–206.
[28] D. L. White, Character degrees of extensions of PSL2(q) and SL2(q), J. Group Theory, 16 (2013) 1–33.
[29] D. L. White, Degree graphs of simple groups, Rocky Mountain J. Math., 39 (2009) 1713–1739.
[30] D. L. White, Degree graphs of simple linear and unitary groups, Comm. Algebra, 34 (2006) 2907–2921.
[31] D. L. White, Degree graphs of simple groups of exceptional Lie type, Comm. Algebra, 32 (2004) 3641–3649.
[32] C. P. M. Zuccari, Regular character degree graphs, J. Algebra, 411 (2014) 215–224.