Finite groups with small character degrees and large prime divisors. II

I. Martin (Irving) Isaacs and Donald Steven Passman

Vol. 29 (1969), No. 2, 311–324

DOI: 10.2140/pjm.1969.29.311

Abstract

In a previous paper one of the authors considered groups G with r.b. n (representation bound n) and n < p² for some prime p. Here we continue this study. We first offer a new proof of the fact that if n = p − 1 then G has a normal Sylow p-subgroup. Next we show that if n = p^3∕2 then p² ∤ |G∕O_p(G)|. Finally we consider n = 2p− 1 and with the help of the modular theory we obtain a fairly precise description of the structure of G. In particular we show that its composition factors are either p-solvable or isomorphic to PSL(2,p), PSL(2,p − 1) for p a Fermat prime or PSL(2,p + 1) for p a Mersenne prime.

Mathematical Subject Classification

Primary: 20.80

Milestones

Received: 24 July 1968

Published: 1 May 1969