# Restrictions on sets of conjugacy class sizes in arithmetic progressions

Document Type : Ischia Group Theory 2020/2021

Authors

1 School of Mathematics, University of East Anglia Norwich, NR4 7TJ, UK

2 Fitzwilliam College, Cambridge, CB3 0DG, UK

Abstract

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 $\alpha=1$.

Keywords

Main Subjects

#### References

[1] R. Baer, Group elements of prime power index, Trans. Amer. Math. Soc., 75 (1953) 20–47.
[2] M. Bianchi, A. Gillio and C. Casolo, A Note on Conjugacy Class Sizes of Finite Groups, Rend. Sem. Mat. Univ. Padova,
106 (2001) 255–260.
[3] 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.
[4] M. Bianchi, S. P. Glasby and Cheryl E. Praeger, Conjugacy class sizes in arithmetic progression, J. Group Theory, 23 no.
6 (2020) 1039–1056.
[5] W. Burnside, Theory of groups of finite order, Cambridge Uuniversity Press, 1911.
[6] A. R. Camina, Arithmetical conditions on the conjugacy class numbers of a finite group, J. London Math. Soc., 2 (1972)
127–132.
[7] A. R. Camina and R. D. Camina, Implications of conjugacy class size, J. Group Theory, 1 no. 3 (1998) 257–269.
[8] A. R. Camina and R. D. Camina, Recognising nilpotent groups, J. Algebra, 300 no. 1 (2006) 16–24.
[9] A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure of finite groups: a survey,
Asian-European J. of Mathematics, 4 no. 4 (2011) 559–588.
[10] J. Cossey and T. Hawkes, Sets of p-powers as conjugacy class sizes, Proc. Amer. Math. Soc., 128 no. 1 (2000) 49–51.
[11] S. Dolfi, On independent sets in the class graph of a finite group, J. Algebra, 303 no. 1 (2006) 216–224.
[12] S. Dolfi and E. Jabara, The structure of finite groups of conjugate rank 2, Bull. London Math. Soc., 41 no. 5 (2009)
219–234.
[13] I. Martin Isaacs, Finite Group Theory, Graduate Studies in Mathematics, 92, American Mathematical Society, Providence, RI, 2008.
[14] K. Ishikawa, On finite p-groups which have only two conjugacy lengths, Israel J. Math., 129 (2002) 119–123.
[15] N. Itô, On finite groups with given conjugate types, I., Nagoya Math. J., 6 (1953) 17–28.
[16] J. Rebmann, F -Gruppen, Arch. Math. (Basel), 22 (1971) 225–230.