Groups with Two Extreme Character Degrees and their Minimal Faithful Representations

Document Type : Original Scientific Paper


Department of Mathematics, Urmia University, Urmia, Iran


for a finite group G, we denote by p(G) the minimal degree of faithful permutation representations of G, and denote by c(G), the minimal degree of faithful representation of G by quasi-permutation matrices over the complex field C. In this paper we will assume that, G is a p-group of exponent p and class 2, where p is prime and cd(G) = {1, |G : Z(G)|1/2}. Then we will show that c(G)≤ |G : Z(G)|1/2 c(Z(G)) , p(G) ≤ |G : Z(G)|1/2p(Z(G)):


Main Subjects

1. H‎. ‎Behravesh‎, ‎Quasi-permutation representations of p-groups of class 2‎, J‎. ‎London Math‎. ‎Soc‎. ‎(2) 55 (2) (1997) 251-260.
2. H‎. ‎Behravesh‎, ‎The minimal degree of a faithful quasi-permutation representation of an abelian group‎, Glasgow Math‎. ‎J. 39 (1) (1997) 51-57.
3. H‎. ‎Behravesh‎, ‎Quasi-permutation representations of metacyclic p-groups‎, ‎ J‎. ‎Sci‎. ‎I‎. ‎R‎. ‎Iran 9 (3) (1998) 258-264‎.
4. H‎. ‎Behravesh and G‎. ‎Ghaffarzadeh‎, ‎Minimal degree of faithful quasi-permutation representations of p-groups‎, ‎Algerba Colloq. 18 (2011) 843-846‎.
5. H‎. ‎Behravesh and H‎. ‎Mousavi‎, ‎A note on p-groups of order p4‎, ‎Proc‎. ‎Indian Acad‎. ‎Sci‎. ‎Math‎. ‎Sci. 119 (2) (2009) 137-143‎.
6. G‎. ‎A‎. ‎Fernandez-Alcober and A‎. ‎Moreto‎, ‎Groups with two extreme character degrees and their normal subgroups‎, ‎Trans‎. ‎Amer‎. ‎Math‎. ‎Soc. 353 (6) (2001) 2171-2192‎.
7. I‎. ‎M‎. ‎Isaacs‎, ‎Character Theory of Finite Groups‎, ‎Academic Press‎, ‎New York‎, ‎1976‎.
8. I‎. ‎M‎. ‎Isaacs and D‎. ‎S‎. ‎Passman‎, ‎A characterization of groups in terms of the degrees of their characters‎, Pacific J‎. ‎Math. 15 (1965) 877-903‎.
9. I‎. ‎M‎. ‎Isaacs and D‎. ‎S‎. ‎Passman‎, ‎A characterization of groups in terms of the degrees of their characters II‎, ‎Pacific J‎. ‎Math. 24 (1968) 467-510‎.
10. D‎. ‎Wright‎, ‎Degree of minimal embeddings for some direct products‎, ‎Amer‎. ‎J‎. ‎Math. 97 (4) (1975) 897-903‎.