Abstract
A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d)log 2 d. Bibliography: 7 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. M. Adel'son-Velskii, B. Yu. Weisfeiler, A. A. Leman, and I. A. Faradjev, “On an example of a graph whose automorphism group is not transitive”,Dokl. Akad. Nauk USSR,185, 975–976 (1969).
C. W. Curtis and I. Reiner,Representation Theory of Finite Groups and Associative Algebras (1962).
S. Evdokimov and I. Ponomarenko, “On isomorphism problem for graphs with bounded multiplicities of spectra,” Research Report No. 85111-CS, University of Bonn (1994).
I. M. Isaacs and D. S. Passman, “Groups with representations of bounded degree,”Canad. J. Math.,16, 299–304 (1964).
W. M. Kantor and E. M. Luks, “Computing in quotient groups,”Proc. 22nd ACM STOC, 524–534 (1990).
B. Weisfeiler, “On the construction and identification of graphs,”Lect. Notes Math.,558 (1976).
H. Wielandt,Finite Permutation Groups, Acad. Press (1964).
Additional information
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119.
Translated by S. A. Evdokimov.
Rights and permissions
About this article
Cite this article
Evdokimov, S.A., Ponomarenko, I.N. Transitive groups with irreducible representations of bounded degree. J Math Sci 87, 4046–4053 (1997). https://doi.org/10.1007/BF02355798
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02355798