A theorem of Hall-Higman type for supersolvable groups

Abstract

Let AC be a finite solvable group with GCIAG and (IAl, [Cl)= 1. Assume P E G is an extraspecial p-subgroup, for some prime p, PcIAG, Z(P) s Z(AG) and A acts faithfully on P/Z(P). Let k be an algebraically closed field of characteristic q #p and let M be a kAG-module such that Z(P) is not trivial on M. In their landmark paper [S], Hall and Higman state some conditions that insure that M(, contains a direct summand isomorphic to kA, in the case where A is cyclic of prime power order. Many authors have extended their results. Notably, Berger [ 11, building on some results of Dade [2], has given some sufficient conditions in the case where A is nilpotent. In [S] the case where char(k) /’ IAl and A is supersolvable is analyzed. In the present paper we look at the case where A is supersolvable and char(k) ) IAl. We prove (Theorem 3.2 below) that, if A does not involve certain groups-essentially wreath products and norm groups (see definitions l.l)-then Ml A contains a direct summand isomorphic to kA. As applications of this result we obtain, in Section 4, some results on modular linear groups with a normal Hall subgroup and a supersolvable complement. For example in Theorem 4.2 we obtain roughly the following result. Let A be a finite supersolvable group which does not involve certain groups. Suppose AC (with GQAG and (IA\, ICI)= 1) is a group of linear transformations of the k-vector space V with 1/l, homogeneous. Then if A acts fixed point freely on V there is some non-trivial normal subgroup of A which centralizes most of G and acts fixed point freely on V. In 1.1 and 4.1, the reader is provided with the necessary definitions for our results. The proof of Theorem 3.2, in the case where char(k) / IAJ, provides a new proof of a slightly weaker version of [S, Theorem 3.41 which does not depend on [S, Theorem 3.21. The results in Section 4 are similar to some nonmodular results in [S]. The reader is given precise directions as to how to adapt their proofs to get these new results. 194 0021-8693/85 $3.00