Conjugacy class numbers and nilpotent  subgroups of finite groups

Hongfei Pan; Shuqin Dong

doi:10.1515/jgth-2023-0263

Published online by De Gruyter April 13, 2024

Conjugacy class numbers and nilpotent subgroups of finite groups

Hongfei Pan and Shuqin Dong

From the journal Journal of Group Theory

https://doi.org/10.1515/jgth-2023-0263

Showing a limited preview of this publication:

Abstract

Let 𝐺 be a finite group, k ⁢ ( G ) the number of conjugacy classes of 𝐺, and 𝐵 a nilpotent subgroup of 𝐺. In this paper, we prove that | B ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | ≤ | G | / k ⁢ ( G ) if 𝐺 is solvable and that 15 7 ⁢ | B ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | ≤ | G | / k ⁢ ( G ) if 𝐺 is nonsolvable, where π = π ⁢ ( B ) is the set of prime divisors of | B | . Both bounds are best possible.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12171058

Award Identifier / Grant number: 12061011

Award Identifier / Grant number: 12201236

Funding source: Natural Science Foundation of Jiangsu Province

Award Identifier / Grant number: BK20231356

Funding statement: The first author was supported by National Natural Science Foundation of China (Nos. 12171058, 12061011) and Natural Science Foundation of Jiangsu Province (No. BK20231356), and the second author by National Natural Science Foundation of China (No. 12201236).

Communicated by: Hung Tong-Viet

References

[1] T. C. Burness, R. Guralnick, A. Moretó and G. Navarro, On the commuting probability of 𝑝-elements in a finite group, Algebra Number Theory 17 (2023), no. 6, 1209–1229. 10.2140/ant.2023.17.1209Search in Google Scholar

[2] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, A ⁢ T ⁢ L ⁢ A ⁢ S of Finite Groups, Clarendon Press, Oxford, 1985. Search in Google Scholar

[3] J. D. Dixon, The Fitting subgroup of a linear solvable group, J. Aust. Math. Soc. 7 (1967), 417–424. 10.1017/S1446788700004353Search in Google Scholar

[4] S. Q. Dong and H. F. Pan, Even character degrees and Ito–Michler theorem, Commun. Math. Stat. (2023), 10.1007/s40304-023-00368-0. 10.1007/s40304-023-00368-0Search in Google Scholar

[5] J. Fulman and R. Guralnick, Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements, Trans. Amer. Math. Soc. 364 (2012), no. 6, 3023–3070. 10.1090/S0002-9947-2012-05427-4Search in Google Scholar

[6] D. Gluck, K. Magaard, U. Riese and P. Schmid, The solution of the k ⁢ ( G ⁢ V ) -problem, J. Algebra 279 (2004), no. 2, 694–719. 10.1016/j.jalgebra.2004.02.027Search in Google Scholar

[7] R. M. Guralnick and G. R. Robinson, On the commuting probability in finite groups, J. Algebra 300 (2006), no. 2, 509–528. 10.1016/j.jalgebra.2005.09.044Search in Google Scholar

[8] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80 (1973), 1031–1034. 10.1080/00029890.1973.11993437Search in Google Scholar

[9] M. W. Liebeck and L. Pyber, Upper bounds for the number of conjugacy classes of a finite group, J. Algebra 198 (1997), no. 2, 538–562. 10.1006/jabr.1997.7158Search in Google Scholar

[10] H. Pan, S. Dong and Y. Yang, Two results on the character degree sums, J. Algebra 584 (2021), 243–259. 10.1016/j.jalgebra.2021.05.015Search in Google Scholar

[11] G. Qian and Y. Yang, The largest conjugacy class size and the nilpotent subgroups of finite groups, J. Group Theory 22 (2019), no. 2, 267–276. 10.1515/jgth-2018-0040Search in Google Scholar

[12] E. P. Vdovin, Large nilpotent subgroups of finite simple groups, Algebra Logic 39 (2000), no. 5, 301–312. 10.1007/BF02681614Search in Google Scholar

[13] The GAP Group, GAP-groups, algorithms, and programming, version 4.10.1, https://www.gap-system.org, 2019. 10.1093/oso/9780190867522.003.0002Search in Google Scholar

Received: 2023-11-22

Revised: 2024-03-12

Published Online: 2024-04-13