Taiwanese Journal of Mathematics

The Influence of Conjugacy Class Sizes on the Structure of Finite Groups

Ruifang Chen and Xianhe Zhao

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Abstract

Let $G$ be a group. The question of how certain arithmetical conditions on the sizes of the conjugacy classes of $G$ influence the group structure has been studied by many authors. In this paper, we investigate the influence of conjugacy class sizes of primary and biprimary elements on the structure of $G$. A criterion for a group to have abelian Sylow subgroups is given and some well-known results on Baer-groups are generalized.

Article information

Source
Taiwanese J. Math., Volume 21, Number 4 (2017), 719-725.

Dates
Received: 3 December 2015
Revised: 2 November 2016
Accepted: 25 November 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501120829

Digital Object Identifier
doi:10.11650/tjm/7083

Mathematical Reviews number (MathSciNet)
MR3684382

Zentralblatt MATH identifier
06871341

Subjects
Primary: 20D25: Special subgroups (Frattini, Fitting, etc.) 20D60: Arithmetic and combinatorial problems

Keywords
conjugacy class size Sylow subgroup abelian

Citation

Chen, Ruifang; Zhao, Xianhe. The Influence of Conjugacy Class Sizes on the Structure of Finite Groups. Taiwanese J. Math. 21 (2017), no. 4, 719--725. doi:10.11650/tjm/7083. https://projecteuclid.org/euclid.twjm/1501120829


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References

  • R. Baer, Group elements of prime power index, Trans. Amer. Math. Soc. 75 (1953), no. 1, 20–47.
  • A. Beltrán and M. José Felipe, Prime powers as conjugacy class lengths of $\pi$-elements, Bull. Austral. Math. Soc. 69 (2004), no. 2, 317–325.
  • W. Burnside, On groups of order $p^{\alpha} q^{\beta}$, Proc. London Math. Soc. s2-1 (1904), no. 1, 388–392.
  • A. R. Camina and R. D. Camina, Implications of conjugacy class size, J. Group Theory 1 (1998), no. 3, 257–269.
  • D. Chillag and M. Herzog, On the length of the conjugacy classes of finite groups, J. Algebra 131 (1990), no. 1, 110–125.
  • B. Fein, W. M. Kantor and M. Schacher, Relative Brauer groups II, J. Reine Angew. Math. 328 (1981), 39–57.
  • X. Guo, X. Zhao and K. P. Shum, On $p$-regular $G$-conjugacy classes and the $p$-structure of normal subgroups, Comm. Algebra 37 (2009), no. 6, 2052–2059.
  • N. Itô, On finite groups with given conjugate types I, Nagoya Math. J. 6 (1953), 17–28.
  • Q. Kong and X. Guo, On an extension of a theorem on conjugacy class sizes, Israel J. Math. 179 (2010), 279–284.
  • X. Zhao and X. Guo, On conjugacy class sizes of the $p'$-elements with prime power order, Algebra Colloq. 16 (2009), no. 4, 541–548.
  • X. H. Zhao, X. Y. Guo and J. Y. Shi, On the conjugacy class sizes of prime power order $\pi$-elements, Southeast Asian Bull. Math. 35 (2011), no. 4, 735–740.