Finite groups with exactly one composite conjugacy class size


A composite number is a positive integer that has at least one divisor integer other than 1 and itself. In this paper, we give a detailed structural description of a group if it has a unique composite conjugacy class size.

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  1. 1.

    Baer R, Group elements of prime power index, Trans. Amer. Math. Soc. 75 (1953) 20–47

    MathSciNet  Article  Google Scholar 

  2. 2.

    Beltrán A and Felipe M J, Normal subgroups and class sizes of elements of prime power order, Proc. Amer. Math. Soc. 140(12) (2012) 4105–4109

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bertram E A, Herzog M and Mann A, On a graph related to conjugacy classes of groups, Bull. London Math. Soc. 22(6) (1990) 569–575

    MathSciNet  Article  Google Scholar 

  4. 4.

    Camina A R, Arithmetical conditions on the conjugacy class numbers of a finite group, J. London Math. Soc. (2) 5 (1972) 127–132

    MathSciNet  Article  Google Scholar 

  5. 5.

    Camina A R and Camina R D, Implications of conjugacy class size, J. Group Theory 1(3) (1998) 257–269

    MathSciNet  Article  Google Scholar 

  6. 6.

    Camina A R and Camina R D, Coprime conjugacy class sizes, Asian-Eur. J. Math. 2(2) (2009) 183–190

    MathSciNet  Article  Google Scholar 

  7. 7.

    Dolfi S and Jabara E, The structure of finite groups of rank 2, Bull. London Math. Soc. 41(5) (2009) 916–926

    MathSciNet  Article  Google Scholar 

  8. 8.

    Itô N, On finite groups with given conjugate types I, Nagoya Math. 6 (1953) 17–28

    MathSciNet  Article  Google Scholar 

  9. 9.

    Jiang Q H and Shao C G, Solvability of finite groups with four conjugacy class sizes of certain elements, Bull. Aust. Math. Soc. 90 (2014) 250–256

    MathSciNet  Article  Google Scholar 

  10. 10.

    Kurzweil H and Stellmacher B, The theory of finite groups: An introduction (2004) (Berlin-Heidelberg-New York: Springer-Verlag)

  11. 11.

    Liu Y J and Liu Y, Finite groups with exactly one composite character degree, J. Algebra Appl. 15(7) (2016) 1650132

    MathSciNet  Article  Google Scholar 

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The authors are thankful to the referee for his/her careful reading and valuable advice. The authors are also grateful to Professor R D Camina who gave them very valuable suggestions. This research was supported by the Natural Science Foundation of Shandong Province (No. ZR2019MA044) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (Numbers 2018QZJ04 and 2017QZJ01).

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Correspondence to Changguo Shao.

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Communicating Editor: Manoj K Yadav

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Jiang, Q., Shao, C. & Zhao, Y. Finite groups with exactly one composite conjugacy class size. Proc Math Sci 130, 5 (2020).

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  • Finite groups
  • conjugacy class sizes
  • quasi-Frobenius groups
  • composite number

2010 Mathematics Subject Classification

  • 20E45
  • 20D15