Zeros of Brauer characters and linear actions of finite groups.
Let \(G\) be a finite group, and \(p\) a prime number greater than \(3\). It is known that, if every irreducible \(p\)-Brauer character of \(G\) does not vanish on any \(p'\)-element of \(G\), then \(G\) is solvable. The primary aim of this work is to describe the structure of groups satisfying the above condition; among other more specific properties, it is shown that the \(p'\)-length of \(G\) is at most 2 (the bound being the best possible). The structural results are obtained as an application of the crucial theorem in this paper, that deals with particular linear actions of solvable groups on finite vector spaces. That theorem runs as follows: Theorem. Let \(G\) be a solvable group, \(q\) a prime number and \(p\) another prime number with \(p\geq 5\). Let \(V\) be a faithful primitive \(G\)-module with \(\#V=q^n\). Assume that \(p\) does not divide the order of \(G\) and that for each orbit for the action of \(G\) on \(V\) there exist an element in such an orbit which is centralized by some \(p'\)-element of \(G\). Then the following conclusions hold: (a) \(G\cong H\), where \(H\leq\Gamma(V)\), the semilinear group acting naturally on \(V\); (b) \(G\) is a Frobenius group with cyclic kernel of order \(p^a\) and Frobenius complement of prime order \(r\) with \(r\) dividing \(n\). Furthermore, \(n\) does not divide \(p-1\) and, for every \(v\in V\setminus\{0\}\), it holds that \(\#C_G(v)=r\). Finally, \(p^a=(q^n-1)/(q^{n/r}-1)\).
scientific article; zbMATH DE number 6005448