Abstract
In 1987, J. G. Thompson put forward the following conjecture: Let G be a finite group with trivial center. If L is a finite simple group satisfying that \(N(G)=N(L)\), then \(G\cong L\). The second author proved above conjecture holds for finite simple groups with non-connected prime graphs. Vasilev proved above conjecture holds for two simple groups with connected prime graphs: \(A_{10}\) and \(L_4(4)\). N. Ahanjideh proved that Thompson’s conjecture is true for \(L_n(q)\). The authors are interested in if it is possible to weaken the conditions in the conjecture. A finite simple group is called a simple \(K_n-\) group if its order is divisible by exactly \(n\) distinct primes. Here, the authors prove that simple \(K_4-\)groups are characterized by their orders and few special conjugacy class sizes, which implies that Thompson’s conjecture is valid for simple \(K_4-\)groups.
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Acknowledgments
The authors would like to thank the referees for carefully reading the earlier version of this paper and making useful suggestions. This work was supported by National Natural Science Foundation of China (Grant Nos.11271301, 11171364, 11001226), the Scientific Research Foundation of Chongqing Municipal Science and Technology Commission (Grant Nos.cstc2014jcyjA00009, cstc2013jcyjA00034), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant Nos. KJ1401006, KJ131204), and the Fundamental Research Funds for the Central Universities.
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Communicated by Kar Ping Shum.
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Chen, Y., Chen, G. & Li, J. Recognizing Simple \(K_4-\)Groups by Few Special Conjugacy Class Sizes. Bull. Malays. Math. Sci. Soc. 38, 51–72 (2015). https://doi.org/10.1007/s40840-014-0003-2
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DOI: https://doi.org/10.1007/s40840-014-0003-2