Restrictions on sets of conjugacy class sizes in arithmetic progressions

Camina, Alan R.; Camina, Rachel D.

doi:10.22108/ijgt.2022.130654.1743

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	Restrictions on sets of conjugacy class sizes in arithmetic progressions
International Journal of Group Theory
مقاله 4، دوره 12، شماره 1، خرداد 2023، صفحه 27-34 اصل مقاله (394.28 K)
نوع مقاله: Ischia Group Theory 2020/2021
شناسه دیجیتال (DOI): 10.22108/ijgt.2022.130654.1743
نویسندگان
Alan R. Camina¹؛ Rachel D. Camina^* ²
¹School of Mathematics, University of East Anglia Norwich, NR4 7TJ, UK
²Fitzwilliam College, Cambridge, CB3 0DG, UK
چکیده
We continue the investigation, that began in [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20--25.] and [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039--1056.], into finite groups whose set of nontrivial conjugacy class sizes form an arithmetic progression. Let $G$ be a finite group and denote the set of conjugacy class sizes of $G$ by ${\rm cs}(G)$. Finite groups satisfying ${\rm cs}(G) = \{1, 2, 4, 6\}$ and $\{1, 2, 4, 6, 8\}$ are classified in [M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no. 6 (2020) 1039--1056.] and [M. Bianchi, A. Gillio and P. P. Pálfy, A note on finite groups in which the conjugacy class sizes form an arithmetic progression, Ischia group theory 2010, World Sci. Publ., Hackensack, NJ (2012) 20--25.], respectively, we demonstrate these examples are rather special by proving the following. There exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha+1}, 2^{\alpha}3 \}$ if and only if $\alpha =1$. Furthermore, there exists a finite group $G$ such that ${\rm cs}(G) = \{1, 2^{\alpha}, 2^{\alpha +1}, 2^{\alpha}3, 2^{\alpha +2}\}$ and $\alpha$ is odd if and only if $\alpha=1$.
کلیدواژه‌ها
conjugacy classes؛ finite soluble groups؛ arithmetic progressions

مراجع
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Restrictions on sets of conjugacy class sizes in arithmetic progressions