The existence of regular orbits

Abstract

A well-known result of Berger (see Theorem 2.2 of [2]) assures that if A is a nilpotent C, C,-free group for all primes p and V is a completely reducible K&module acted faithfully by A, then V contains a regular A-orbit, i.e., there exists v E V with C,(v) = 1. This result was improved by Hargraves in [7], showing that it suffices to assume that A is C, 1 C,-free for p = 2 and p a Mersenne prime. Her proof has been recently simplified by Fleischmann in [4]. Berger’s result has been improved in a different direction by Turull in [lo], who extended it to the supersolvable case with extra assumptions. If A is nilpotent and both char(K) and IAl are odd, then the result holds without any restriction as Gow showed in [S]. All these results assume the complete reducibility of V. In fact, if A is nilpotent, they suppose that char(K) t IAl. The purpose of this note is to consider the modular case in an important particular situation. We prove that if A is a p-subgroup of a solvable group G and H/K is a p-chief factor of G acted faithfully by A, then H/K contains a regular A-orbit provided that A is C2 2 C,-free when p = 2. Combining this with the theorems of Hargraves and Gow we obtain several corollaries. One of them is the following: In a group of odd order every p-subgroup has a regular orbit on each chief factor on which it acts faithfully. Easily constructed examples show that the analogues of the Berger and Gow theorems are false when char(K) 1 JAI. Thus some restriction seems to be necessary. We do not know if our theorem is true when p-group is replaced by nilpotent. We give three applications of our theorem. The first is about fixed point free automorphism groups. A well-known situation is the following: Suppose AC is a solvable group, GaAG, ([AI, IGl)=l, V is a KAG-module, char(K) 1 IAl, and C,(A) =O. Then, if V is not a primitive KAG-module, we may obtain a nontrivial subgroup of A centralizing a section of G (see Proposition 4.1 of [IO]). Nothing seems to be known when (IA 1, ICI) # 1. We obtain a weak analogue of Turull’s result in our Proposition 2.1. The main obstruction to obtain a quite similar result in