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 \({v_1, v_2 \in V}\) such that \({{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P}\) , where \({P \in {\rm Syl}_p(G)}\) . We hence deduce that, if the normal p-complement K is nontrivial, there exists \({v \in {\bf C}_{V}(P)}\) such that |K : C K (v)|2 > |K|.
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S. Dolfi is partially supported by MURST project “Teoria dei Gruppi e Applicazioni”.
G. Navarro is partially supported by the Ministerio de Educación y Ciencia proyecto MTM-2007-61161.
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Dolfi, S., Navarro, G. Large orbits of elements centralized by a Sylow subgroup. Arch. Math. 93, 299–304 (2009). https://doi.org/10.1007/s00013-009-0016-5
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DOI: https://doi.org/10.1007/s00013-009-0016-5