The largest orbit sizes of linear group actions and abelian quotients (Q6564594)

Let \(G\) be a finite group and let \(V\) be a finite and faithful \(G\)-module such that \(V\) is completely reducible, possibly of mixed characteristic, or \(V\) is a \(G\)-module over a field of characteristic \(p\) with \(O_{p}(G) =1\) and let \(M\) be the largest orbit size in the action of \(G\) on \(V\). The second author and \textit{Y. Yang} [Isr. J. Math. 211, 23-44 (2016; Zbl 1353.20007)] showed that if \(G\) is soluble, then \(|G:G'| \leq M\) and they later extended this bound to arbitrary groups [Sci. China, Math. 63, No. 8, 1523--1534 (2020; Zbl 1472.20021)].\N\NIn this context, an interesting problem is to classify all linear group actions in the case \(|G :G'|=M\). Let \(V=V(n,q)\) be the \(n\)-dimensional vector space over the field with \(q\) elements. The main result of this paper is (Theorem A): \N\NLet \(G\) be a finite nonabelian group and let \(V\) be a finite faithful irreducible \(G\)-module. If \(|G:G'|=M\), then one of the following situations occurs. \N(a) \(G\) is \(D_{8}\) and \(V=V(2,3)\); \N(b) \(G\) is the central product of \(D_{8}\) and \(C_{4}\) and \(V=V (2,5)\); \N(c) \(G\) is the central product of \(D_{8}\) and \(Q_{8}\) and \(V=V (4,3)\).\N\NThe authors also include an application of the main results to Brauer's \(k(B)\)-problem.