This paper studies finite linear groups of relatively small degree over I~, the real numbers, and Q, the rationals. Throughout the paper, G denotes a finite group, p a fixed rational prime which divides IG[, P a Sylow p-subgroup of G, and 1 ~ the trivial character for G over a field of characteristic zero.
If G has a unique composition factor (with multiplicity one) whose order is divisible by p; we denote it by CFp(G). When P is cyclic, it is a wellknown consequence of Burnside's transfer theorem (see, for example, r l, Lemma 5
.1]) that OP'(G)Op,(G)/Op,(G)(~-OP'(G)/OP'(G)c~Op,(G)) is either isomorphic to P or is a nonabelian simple group with P~OP'(G)/O"(G)c~Or.(G). Thus, G is not p-solvable if and only if CFp(G) ~-OP'(G)/OP'(G) c~ Op.(G) ~ P if and only if OP'(G) is perfect.
The following definition is due to Feit 1,8].
DEFINITION. Assume that P is cyclic. G is said to be of type L2(p) if OP'(G)/OP'(G) c~ Op,(G) is isomorphic to either P or PSL(2, p).
THEOREM A. Assume that P is cyclic and that X is a faitl~d representation of G over afield F, with F~ R. Let Z be the character of X and assume also that Z(I ) ~< IPI -1. Then one of the followhlg must hold: IPI and either G'~SL(2, 2m)โข with 2"'+I=IPI=(p or 9)~<7.(1)+2, or 7.=(1~-1~)2, where 2 is a lhlear character of G with 22 = I c (so that X is equivalent to a representation over Q); or