Large orbits of elements centralized by a Sylow subgroup

摘要

If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist v₁, v₂ Î V{v_1, v_2 in V} such that C_G(v₁)ÇC_G(v₂) = P{{bf C}_{G}{(v_1)}cap {bf C}_{G}{(v_2)} = P} , where P Î Syl_p(G){P in {rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v Î C_V(P){v in {bf C}_{V}(P)} such that |K : C _K (v)|2 > |K|.