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ON $p$ -PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS

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

Abstract

Let $G$ be a finite group. Let $\operatorname{cl}(G)$ be the set of conjugacy classes of $G$ and let $\operatorname{ecl}_{p}(G)$ be the largest integer such that $p^{\operatorname{ecl}_{p}(G)}$ divides $|C|$ for some $C\in \operatorname{cl}(G)$ . We prove the following results. If $\operatorname{ecl}_{p}(G)=1$ , then $|G:F(G)|_{p}\leq p^{4}$ if $p\geq 3$ . Moreover, if $G$ is solvable, then $|G:F(G)|_{p}\leq p^{2}$ .

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The project was supported by NSFC (Nos. 11671063 and 11471054), the Natural Science Foundation Project of CSTC (cstc2016jcyjA0065) and the NSF of Jiangsu Province (No. BK20161265). The first author was also supported by a grant from the Simons Foundation (No. 499532).

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[1] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of Finite Groups (Oxford University Press, New York, 1985).
[2] Lewis, M., Navarro, G., Tiep, P. H. and Tong-Viet, H. P., ‘ p-parts of character degrees’, J. Lond. Math. Soc. (2) 92 (2015), 483497.
[3] Lewis, M., Navarro, G. and Wolf, T. R., ‘ p-parts of character degrees and the index of the Fitting subgroup’, J. Algebra 411 (2014), 182190.
[4] Liu, X., Wang, Y. and Wei, H., ‘Notes on the length of conjugacy classes of finite groups’, J. Pure Appl. Algebra 196 (2005), 111117.
[5] Michler, G., ‘A finite simple group of Lie type has p-blocks with different defects, p≠2’, J. Algebra 104(2) (1986), 220230.
[6] Qian, G. and Wang, Y., ‘A note on the conjugacy class sizes of finite groups’, Acta Math. Sci. Chin. Ser. 52(1) (2009), 125130.
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Keywords

MSC classification

ON $p$ -PARTS OF CONJUGACY CLASS SIZES OF FINITE GROUPS

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

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