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ON SUB-CLASS SIZES OF FINITE GROUPS

  • GUOHUA QIAN (a1) and YONG YANG (a2) (a3)

Abstract

For every element $x$ of a finite group $G$ , there always exists a unique minimal subnormal subgroup, say, $G_{x}$ of $G$ such that $x\in G_{x}$ . The sub-class of $G$ in which $x$ lies is defined by $\{x^{g}\mid g\in G_{x}\}$ . The aim of this paper is to investigate the influence of the sub-class sizes on the structure of finite groups.

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Copyright

Corresponding author

Footnotes

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Email addresses for correspondence: email ghqian2000@163.com, yang@txstate.edu.

This project was supported by the NSF of China (nos. 11471054, 11671063, and 11871011), the NSF of Jiangsu Province (no. BK20161265), the Natural Science Foundation of Chongqing (cstc2016jcyjA0065, cstc2018jcyjAX0060), and a grant from the Simons Foundation (no. 499532).

Footnotes

References

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[1]Camina, A. R., ‘Arithmetical conditions on the conjugacy class numbers of a finite group’, J. Lond. Math. Soc. 5 (1972), 127132.
[2]Camina, A. R. and Camina, R. D., ‘The influence of conjugacy class sizes on the structure of finite groups: a survey’, Asian-Eur. J. Math. 4 (2011), 559588.
[3]Casolo, C. and Dolfi, S., ‘Conjugacy class lengths of metanilpotent groups’, Rend. Semin. Mat. Univ. Padova 96 (1996), 121130.
[4]Casolo, C. and Dolfi, S., ‘Prime divisors of irreducible character degrees and of conjugacy class sizes in finite groups’, J. Group Theory 10 (2007), 571583.
[5]Chillag, D. and Herzog, M., ‘On the length of the congugacy classes of finite groups’, J. Algebra 131 (1990), 110125.
[6]Isaacs, I. M., Finite Group Theory, Graduate Studies in Mathematics, 92 (American Mathematical Society, Providence, RI, 2008).
[7]Itô, N., ‘On finite groups with given conjugate types. I’, Nagoya Math. J. 6 (1953), 1728.
[8]Huppert, B., Character Theory of Finite Groups (Walter de Gruyter, Berlin, 1998).
[9]Kurzweil, H. and Stellmacher, B., The Theory of Finite Groups (Springer, New York, 2004).
[10]Liu, X., Wang, Y. and Wei, H., ‘Notes on the length of conjugacy classes of finite groups’, J. Pure Appl. Algebra 196 (2005), 111117.
[11]Zhang, J., ‘On the lengths of conjugacy classes’, Comm. Algebra 26 (1998), 23952400.
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ON SUB-CLASS SIZES OF FINITE GROUPS

  • GUOHUA QIAN (a1) and YONG YANG (a2) (a3)

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