Finite simple groups with number of zeros slightly greater than the number of nonlinear irreducible characters

Document Type: Research Paper

Author

the Chinese Mathematical Society

Abstract

The aim of this paper is to classify the finite simple groups with‎ ‎the number of zeros at most seven greater than the number of‎ ‎nonlinear irreducible characters in the character tables‎. ‎We find‎ ‎that they are exactly A$_{5}$‎, ‎L$_{2}(7)$ and A$_{6}$‎.

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Y. Berkovich and L. Kazarin (1998). Finite groups in which the zeros of every nonlinear irreducible character are conjugate modulo its kernel. Houston J. Math.. 24 (4), 619-630
R. T. Curtis, S. P. Norton, R. A. Park and R. A. Wilson (1985). Atlas of Finite Groups. Oxford Univ. Press (Clarendon), New York.
H. W. Deng and W. J. Shi (1997). A simplicity criterion for finite groups. J. Algebra. 191, 371-381
X. L. Du and W. J. Shi (2006). Finite groups with conjugacy classes number one greater than its same order classes number. Comm. Algebra. 34, 1345-1359
T. L. Huang and W. J. Shi (1995). Finite groups all of whose element orders are of prime power except one (in Chinese). J. of the Southwest China Normal University. 20, 610-617
B. Huppert (1982). Finite Groups I. Springer, Heidelberg.
B. Huppert and N. Blackburn (1982). Finite Groups III. Springer, Berlin.
N. Iiyori and H. Yamaki (1993). Prime graph components of the simple groups of Lie type over the field of even characteristic. J. Algebra. 155, 335-343
I. M. Isaacs (1976). Character Theory of Finite Groups. Academic Press, New York.
A. Jafarzadeh and A. Iranmanesh (2007). On simple K$_{n}$-groups for $n=56$. Groups St Andrews 2005 Vol. 2 (Edited by C.M. Campbell, M. R. Quick, E. F. Robertson and G. C. Smith), London Math. Soc. Lecture Note Ser. Cambridge Univ.. 340, 517-526
J. McKay (1979). The non-abelian simple groups $G$, $|G|< 10^{6}$-character tables. Comm. Algebra. 7, 1407-1445
A. Moret$acute{o}$ and J. Sangroniz (2004). On the number of conjugacy classes of zeros of characters. Israel J. Math.. 142, 163-187
M. Suzuki (1962). On a class of doubly transitive groups. Ann. of Math. (2). 75, 105-145
M. Suzuki (1961). Finite groups with nilpotent centralizers. Trans. Amer. Math. Soc.. 99, 425-470
A. Veralopez and J. Veralopez (1986). Classification of finite groups according to the number of conjugacy classes II. Israel J. Math.. 56, 188-221
W. Willems (1988). Blocks of defect zero in finite simple groups. J. Algebra. 113, 511-522
J. S. Williams (1981). Prime graph components of finite groups. J. Algebra. 69, 487-513
J. S. Zhang, T. J. Shi and Z. C. Shen (2010). Finite groups in which every irreducible character vanishes on at most three conjugacy classes. J. Group Theory. 13, 799-819