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On the centralizers of the p-regular elements in a finite group

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Abstract

Let G be a p-solvable finite group for some prime p, \(G_{p'}\) a \(p'\)-Hall subgroup of G and x a p-regular element of G. Clearly, \(\langle x\rangle \le C_G(x)\le G\). Notice that the structure of \(G_{p'}\) is easily decided when \(C_G(x)=\langle x\rangle \) and \(C_G(x)=G\) for every p-regular element x of G. So, in this paper, we investigate the structure of \(G_{p'}\) by assuming that \(C_G(x)\) is maximal in G for every non-central p-regular element x of G.

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References

  • Ashrafi, A.R.: On finite groups with a given number of centralizers. In: Algebra Colloquium, vol. 7, pp. 139–146. Springer (2000)

  • Babai, L., Pálfy, P.P., Saxl, J.: On the number of \(p-\)regular elements in finite simple groups. LMS J. Comput. Math. 12, 82–119 (2009)

    Article  MathSciNet  Google Scholar 

  • Beltrán, A., Felipe, M.J.: Certain relations between \(p-\)regular class sizes and the \(p-\)structure of \(p-\)solvable groups. J. Aust. Math. Soc. 77(3), 387–400 (2004)

    Article  MathSciNet  Google Scholar 

  • Berkovich, Y., Kazarin, L.: Indices of elements and normal structure of finite groups. J. Algebra 283(2), 564–583 (2005)

    Article  MathSciNet  Google Scholar 

  • Doerk, K., Hawkes, T.O.: Finite Soluble Groups, vol. 4. Walter de Gruyter, Berlin (2011)

    MATH  Google Scholar 

  • Fein, B., Kantor, W.M., Schacher, M.: Relative Brauer groups ii. J. Reine Angew. Math. 328, 39–57 (1981)

    MathSciNet  MATH  Google Scholar 

  • Feit, W., Hall, M., Thompson, J.G.: Finite groups in which the centralizer of any non-identity element is nilpotent. Math. Zeits. 74(1), 1–17 (1960)

    Article  MathSciNet  Google Scholar 

  • Jafarian Amiri, S.M., Amiri, M., Rostami, H.: Finite groups determined by the number of element centralizers. Commun. Algebra 45(9), 3792–3797 (2017)

    Article  MathSciNet  Google Scholar 

  • Jafarian Amiri, S.M., Rostami, H.: Finite groups in which the centralizer of every noncentral element of odd order is abelian. J. Algebra Appl. 18(06), 1950108 (2019)

    Article  MathSciNet  Google Scholar 

  • Michio, S.: Group Theory II. Springer, New York (1986)

    MATH  Google Scholar 

  • Navarro, G., Tiep, P.H.: Abelian sylow subgroups in a finite group. J. Algebra 398, 519–526 (2014)

    Article  MathSciNet  Google Scholar 

  • Robinson, D.J.: A Course in the Theory of Groups, vol. 80. Springer Science & Business Media, New York (2012)

    Google Scholar 

  • Yang, Y., Qian, G.: On \( p \)-parts of conjugacy class sizes of finite groups. Bull. Aust. Math. Soc. 97(3), 406–411 (2018)

    Article  MathSciNet  Google Scholar 

  • Zarrin, M.: On element-centralizers in finite groups. Arch. Math. 93(6), 497 (2009)

    Article  MathSciNet  Google Scholar 

  • Zhao, X., Guo, X., Shi, J.: On the conjugacy class sizes of prime power order \(\pi \)-elements. Southeast Asian Bull. Math. 35(4), 89 (2011)

    MathSciNet  Google Scholar 

  • Zhao, X., Chen, R., Geng, X.: On \(p-\)regular \({G}\)-conjugacy class sizes. J. Inequalities Appl. 2014(1), 34 (2014)

    Article  MathSciNet  Google Scholar 

  • Zhao, X., Chen, R., Guo, X.: Groups in which the centralizer of any non-central element is maximal. J. Group Theory (2020). https://doi.org/10.1515/jgth-2019-0150

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for their valuable suggestions and useful comments contributed to the final version of this paper.

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Correspondence to Xianhe Zhao.

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The research of the work was supported by the National Natural Science Foundation of China (11501176, 11901169), the project for high quality courses of postgraduate education in Henan Province, Research and practice project of higher education reform in Henan Normal University (post-graduate education, No. YJS2019JG06) and Key Laboratory of Applied Mathematics of Fujian Province University (Putian University, No. SX201902).

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Zhao, X., Chen, R., Zhou, Y. et al. On the centralizers of the p-regular elements in a finite group. Bull Braz Math Soc, New Series 52, 353–360 (2021). https://doi.org/10.1007/s00574-020-00207-8

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  • DOI: https://doi.org/10.1007/s00574-020-00207-8

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