Certain finite linear groups of prime degree

Abstract

In studying finite linear groups of fixed degree over the complex field, it is convenient to restrict attention to irreducible, unimodular, and quasiprimitive groups. If one assumes the degree to be an odd prime p, there is a natural division into cases, according to the order of a Sylow p-group of such a group. When the order is p 4 or larger, all such groups are known (by W. Feit and J. Lindsey, independently). THEOREM 1 . Suppose G is a finite group with a faithful, irreducible, unimodular, and quasiprimitive complex representation of prime degree p ≥ 5. If a Sylow p-group P of G has order p 3 , then P is normal in G. As is well known, Theorem 1 is false for p = 2 or 3. Combining Theorem 1 with known results, we have immediately the following conjecture of Feit. THEOREM 2 . Suppose G is a finite group with a faithful, irreducible, and unimodular complex representation of prime degree p ≥ 5. Then p 2 does not divide the order of G/0 p (G). The following result, which is of independent interest, is used in the proof of Theorem 1. THEOREM 3 . Suppose G is a finite group with a Sylow p-group P of order larger than 3, which satisfies C G (x) = P, for all x ≠ 1 in P. If G has a faithful complex representation of degree less than (|P| - 1) 2/3 , then P is normal in G.