Large orbits of elements centralized by a Sylow subgroup |
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Authors: | Silvio Dolfi Gabriel Navarro |
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Institution: | 1. Dipartimento di Matematica U. Dini, viale Morgagni, 67/a, 50134, Firenze, Italy 2. Departament d’ àlgebra, Facultat de Matemàtiques, Universitat de València, 46100, Burjassot, València, Spain
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Abstract: | 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 ${v_1, v_2 \in V}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 ? V{v_1, v_2 \in V} such that CG(v1)?CG(v2) = P{{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P} , where P ? Sylp(G){P \in {\rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v ? CV(P){v \in {\bf C}_{V}(P)} such that |K : C
K
(v)|2 > |K|. |
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