On finite groups with square-free conjugacy class sizes

Document Type: Ischia Group Theory 2016

Authors

Universidad Politecnica de Valencia

Abstract

We report on fi nite groups having square-free conjugacy class sizes, in particular in the framework of factorised groups.

Keywords

Main Subjects


[1] M. Asaad and A. Shaalan, On the supersolvability of finite groups, Arch. Math. (Basel), 53 (1989) 318-326.

[2] A. Ballester-Bolinches, J. Cossey and Y. Li, Mutually permutable products and conjugacy classes, Monatsh. Math., 170 (2012) 305–310.

[3] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of finite groups, de Gruyter Expositions in Mathematics, 53, Walter de Gruyter GmbH & Co. KG, Berlin, 2010.

[4] A. Beltrán and C. G. Shao, Coprime action and arithmetical conditions on invariant conjugacy classes, Sci. China Math., 58 (2015) 2499–2504.

[5] J. Brough, On vanishing criteria that control finite group structure, J. Algebra, 458 (2016) 207–215.

[6] A. R. Camina and R. D. Camina, The influence of conjugacy class sizes on the structure of finite groups: a survey, Asian-Eur. J. Math., 4 (2011) 559–588.

[7] A. R. Camina and R. D. Camina, Implications of conjugacy class size, J. Group Theory, 1 (1998) 257–269.

[8] D. Chillag and M. Herzog, On the length of the conjugacy classes of finite groups, J. Algebra, 131 (1990) 110–125.

[9] J. Cossey and Y. Wang, Remarks on the length of conjugacy classes of finite groups, Comm. Algebra, 27 (1999) 4347–4353.

[10] K. Doerk and T. Hawkes, Finite Soluble Groups, de Gruyter Expositions in Mathematics, 4 Walter de Gruyter GmbH & Co. KG, Berlin, 1992.

[11] M. J. Felipe, A. Mart´─▒nez-Pastor and V. M. Ortiz-Sotomayor, Square-free class sizes in products of groups, submitted.

[12] H. G. Knoche,Über den Frobeniusschen Klassenbegriff in nilpotenten Gruppen, Math. Z., 55 (1951) 71–83.

[13] L. M. Lewis, G. Navarro, P. H. Tiep and H. P. Tong-Viet, p-parts of character degrees, J. Lond. Math. Soc. (2), 92 (2015) 483–497.

[14] L. M. Lewis, G. Navarro and T. R. Wolf, p-parts of character degrees and the index of the Fitting subgroup, J. Algebra, 411 (2014) 182–190.

[15] S. Li, On the class length of elements of prime power order in finite groups, Guangxi Sci., 6 no. 1 (1999) 12–13.

[16] X. Liu, Y. Wang and H. Wei, Notes on the length of conjugacy classes of finite groups, J. Pure Appl. Algebra, 196 (2005) 111–117.

[17] G. Qian and Y. Wang, On conjugacy class sizes and character degrees of finite groups, J. Algebra Appl., 13 (2014) 13501001–13501009.

[18] X. Shi, H. Wei and X. Ma, Lengths of conjugacy classes of factorized groups and structures of finite groups (Chinese), J. Zhengzhou Univ. Nat. Sci. Ed., 45 no. 2 (2013) 10–13.

[19] The GAP Group, GAP–Groups, Algorithms, and Programming, Version 4.8.4 (June 2016).