Finite groups with non-trivial intersections of kernels of all but one irreducible characters

Document Type : Ischia Group Theory 2016


1 Dipartimento di Matematica quot;Federigo Enriques quot;, Università di Milano

2 Schoool of Mathematical Sciences, Tel-Aviv University


In this paper we consider finite groups $G$ satisfying the following‎ ‎condition‎: ‎$G$ has two columns in its character table which differ by exactly one‎ ‎entry‎. ‎It turns out that such groups exist and they are exactly the finite groups‎ ‎with a non-trivial intersection of the kernels of all but one irreducible‎ ‎characters or‎, ‎equivalently‎, ‎finite groups with an irreducible character‎ ‎vanishing on all but two conjugacy classes‎. ‎We investigate such groups‎ ‎and in particular we characterize their subclass‎, ‎which properly contains‎ ‎all finite groups with non-linear characters of distinct degrees‎, ‎which were characterized by Berkovich‎, ‎Chillag and Herzog in 1992‎.


Main Subjects

[1] Ya. Berkovich, D. Chillag and M. Herzog, Finite groups in which the degrees of the nonlinear irreducible characters are distinct, Proc. Amer. Math. Soc., 115 (1992) 955–959.
[2] Ya. Berkovich, I. M. Isaacs and L. Kazarin, Groups with distinct monolithic character degrees, J. Algebra, 216 (1999) 448–480.
[3] S. Dolfi, G. Navarro and P. H. Tiep, Finite groups whose same degree characters are Galois conjugate, Israel J. Math., 198 (2013) 283–331.
[4] S. Dolfi and M. K. Yadav, Finite groups whose non-linear irreducible characters of the same degree are Galois conjugate, J. Algebra, 452 (2016) 1–16.
[5] W. Feit, Characters of finite groups, W. A. Benjamin Inc., New York-Amsterdam, 1967.
[6] S. S. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math., 109 (1983) 363–385.
[7] M. Hall, The theory of groups, AMS Chelsea Publishing, Providence, Rhode Island, 1999.
[8] B. Huppert, Character theory of finite groups, Walter de Gruyter, Berlin, 1998.
[9] M. Loukaki, On distinct character degrees, Israel J. Math., 159 (2007) 93–107.
[10] D. Passman, Permutation groups, W. A. Benjamin Inc., New York-Amsterdam, 1968.
[11] G. Seitz, Finite groups having only one irreducible representation of degree greater than one, Proc. Amer. Math. Soc., 19 (1968) 459–461.
[12] H. Zassenhaus, Kennzeichnung endlicher linear Gruppen als Permutationsgruppen, Hamburg Abh., 11 (1936) 17–40.
Volume 7, Issue 3 - Serial Number 3
Proceedings of the Ischia Group Theory 2016-Part III
September 2018
Pages 63-80
  • Receive Date: 15 July 2016
  • Revise Date: 23 March 2017
  • Accept Date: 17 March 2017
  • Published Online: 01 September 2018