Large orbits of elements centralized by a Sylow subgroup.
Let \(G\) be a finite group. In this paper, the authors prove that if \(G\) has a nilpotent normal \(p\)-complement and \(V\) is a finite, faithful and completely reducible \(G\)-module of characteristic \(p\), then there exist \(v_1,v_2 \in V\) such that \(C_G(v_1)\cap C_G(v_2)=P\), where \(P\in\text{Syl}_p(G)\). An interesting corollary is: if the normal \(p\)-complement \(K\) is non-trivial, there exists \(v\in C_V(P)\) such that \(|K:C_K(v)|^2>|K|\).
scientific article; zbMATH DE number 5638426