Recognition of Janko groups and some simple $K_4$-groups by the order and one irreducible character degree or character degree graph

Document Type : Research Paper

Authors

1 Department of Mathematics, Urmia University, Urmia , Iran

2 Department of Mathematics, Khoy Branch, Islamic Azad University, Khoy , Iran

Abstract

‎In this paper we prove that some Janko groups are uniquely‎ ‎determined by their orders and one irreducible character‎ ‎degree‎. ‎Also we prove that some finite simple $K_4$-groups are‎ ‎uniquely determined by their character degree graphs and their‎ ‎orders‎.

Keywords

Main Subjects


[1] R. W. Carter, Finite Groups of Lie type: conjugacy classes and complex characters, John Wiley and Sons, Chichester,
England, 1985.
[2] 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.
[3] B. Huppert, Some simple groups which are determined by the set of their character degrees I, Illinois J. Math., 44
(2000) 828–842.
[4] B. Huppert, Some simple groups which are determined by the set of their character degrees II, Rend. Semin. Mat.
Univ. Padova, 115 (2006) 1–13.
[5] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, San Diego, California, 1976.
[6] B. Khosravi, B. Khosravi, B. Khosravi and Z. Momen, Recognition by character degree graph and order of simple
groups of order less than 6000, Miskolc Math. Notes, 15 (2014) 537–544.
[7] B. Khosravi, B. Khosravi, B. Khosravi, Z. Momen, Recognition of the simple group PSL(2, p2) by character degree
graph and order, Monatsh Math., 178 (2015) 251–257.
[8] B. Khosravi, B. Khosravi, B. Khosravi and Z. Momen, Recognition of some simple groups by character degree graph
and order, Math. Reports, 18 (2016) 51–61.
[9] M. L. Lewis, An overview of graphs assosiated with character degrees and conjugacy class sizes in finite groups,
Rocky Mountain J. Math., 38 (2008) 175–211.
[10] O. Manz, R. Staszewski and W. Willems, On the number of components of a graph related to character degrees,
Proc. Amer. Math. Soc., 103 (1988) 31–37.
[11] A. Moret´o and P. H. Tiep, Prime divisors of character degrees, J. Group Theory, 11 (2008) 341–356.
[12] P. P´alfy, On the character degree graph of solvable groups. I. Three primes, Period. Math. Hungar., 36 (1998) 61–65.
[13] T. P. Wakefield, Verifying Huppert’s conjecture for PSL3(q) and PSU3(q2), Commun. Algebra, 37 (2009) 2887–2906.
[14] D. L. White, Degree graphs of simple groups, Rocky Mountain J. Math., 39 (2009) 1713–1739.
[15] H. P. Tong and T. P. Wakefield, On Huppert’s conjecture for G2(q), q ≥ 7, J. Pure Appl. Algebra, 216 (2012)
2720–2729.
[16] H. Xu, Y. Chen and Y. Yan, A new characterization of simple K3-group by their orders and large degrees of their
irreducible characters, Comm. Algebra, 42 (2014) 5374–5380.
[17] H. Xu, Y. Chen and Y. Yan, A new characterization of Mathieu-groups by the order and one irreducible character
degree, J. Ineq. App., 209 (2013) 1–6.