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Simple groups and the two-prime hypothesis

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Abstract

Let G be a finite group, and write \(\mathrm{cd}(G)\) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if, for any distinct degrees \(a, b \in \mathrm{cd}(G)\), the total number of (not necessarily different) primes of the greatest common divisor \(\gcd (a, b)\) is at most 2. In this paper, we determine the nonabelian chief factors that can occur for groups satisfying the two-prime hypothesis.

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Acknowledgments

This project was started while both authors took part in the “Third International Symposium on Groups, Algebras and Related topics in Beijing, celebrating the 50th Anniversary of the Journal of Algebra”. The authors would like to express their deep thanks to the Beijing International Center for Mathematical Research, and particularly Professor Jiping Zhang, for encouraging them to get together to focus on the topic of this paper. The second author was partially supported by the National Natural Science Foundation of China (11201194) and (11471054) and the Science Foundation of Jiangxi Province for Youths (20142BAB211011). Finally, the authors are deeply indebted to the anonymous referee from Algebra and Representation Theory who carefully read our previous long manuscript and made many corrections and suggestions. This leads to the present four-part division of the whole work.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Kent State University, Kent, OH, 44242, USA

    Mark L. Lewis

  2. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, 330022, People’s Republic of China

    Yanjun Liu

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  1. Mark L. Lewis
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  2. Yanjun Liu
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Correspondence to Mark L. Lewis.

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Communicated by J. S. Wilson.

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Lewis, M.L., Liu, Y. Simple groups and the two-prime hypothesis. Monatsh Math 181, 855–867 (2016). https://doi.org/10.1007/s00605-015-0839-z

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  • DOI: https://doi.org/10.1007/s00605-015-0839-z

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