FINITE GROUPS WITH SMALL CHARACTER DEGREES AND LARGE PRIME DIVISORS .2.

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 psubgroup. Next we show that if n = p3/2 then p2 |G/Op(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. © 1969 by Pacific Journal of Mathematics.