Bounding the even abelian composition factors in finite linear groups (Q6117064)

Let \(G\) be a finite group acting faithfully and completely reducibly on a vector space \(V\), let \(\xi_{2}(G)\) be the product of the orders of the even abelian composition factors of \(G\) and let \(\beta = \log(32)/\log(9)\). The main result of this paper is the inequality \(\xi_{2}(G) \leq |V|^{\beta}/2\). The proof is based on the methods exposed in the monograph [\textit{O. Manz} and \textit{T. R. Wolf}, Representations of solvable groups. Cambridge: Cambridge University Press (1993; Zbl 0928.20008)].