Abstract
An element of a group G is called semi-rational if all generators of \(\langle x\rangle \) lie in the union of two conjugacy classes of G. If all elements of G are semi-rational, then G is called a semi-rational group. In this paper, we determine all semi-rational simple groups. Our study in this article generalises Feit and Seitz’s result (Ill J Math 33(1):103–131, 1989) to semi-rational groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alavi, S.H., Daneshkhah, A., Darafsheh, M.R.: On semi-rational Frobenius groups. J. Algebra Appl. 15(02), 1650,033 (2016). doi:10.1142/S021949881650033X
Babai, L., Pálfy, P.P., Saxl, J.: On the number of \(p\)-regular elements in finite simple groups. LMS J. Comput. Math. 12, 82–119 (2009). doi:10.1112/S1461157000000036
Chillag, D., Dolfi, S.: Semi-rational solvable groups. J. Group Theory 13(4), 535–548 (2010). doi:10.1515/JGT.2010.004
Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A., Wilson, R.A.: Atlas of Finite Groups. Oxford University Press, Eynsham (1985). Maximal subgroups and ordinary characters for simple groups. With computational assistance from J. G. Thackray
Feit, W., Seitz, G.M.: On finite rational groups and related topics. Ill. J. Math. 33(1), 103–131 (1989). http://projecteuclid.org/getRecord?id=euclid.ijm/1255988808
Fleischmann, P., Janiszczak, I.: The semisimple conjugacy classes of finite groups of Lie type \(E_6\) and \(E_7\). Commun. Algebra 21(1), 93–161 (1993). doi:10.1080/00927879208824553
The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.8.2 (2016). http://www.gap-system.org
Gow, R.: Groups whose characters are rational-valued. J. Algebra 40(1), 280–299 (1976)
Kleidman, P.B., Liebeck, M.W.: The Subgroup Structure of the Finite Classical Groups, London Mathematical Society Lecture Note Series, vol. 129. Cambridge University Press, Cambridge (1990)
Acknowledgments
The authors are grateful to the anonymous referees and the editor for careful reading of the manuscript and for corrections and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
Rights and permissions
About this article
Cite this article
Alavi, S.H., Daneshkhah, A. On semi-rational finite simple groups. Monatsh Math 184, 175–184 (2017). https://doi.org/10.1007/s00605-016-0964-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-016-0964-3