On Certain Distributive Lattices of Subgroups of Finite Soluble Groups  

作　　者：L.M.EZQUERRO X.SOLER-ESCRIV 

机构地区：[1]InformáticaUniversidad Pública de Navarra,Campus de Arrosadía.31006 Pamplona,Spain [2]Departamentde Matemàtica Aplicada,Universitat d'Alacant,Campus de Sant Vicent.Ap.Correus 99-03080 Alacant,Spain

出　　处：《Acta Mathematica Sinica,English Series》2007年第11期2069-2078,共10页数学学报（英文版）

基　　金：Proyecto BFM 2001-1667-C03-01

摘　　要：In this paper, we prove the following result. Let ξ be a saturated formation and ∑ a Hall system of a soluble group G. Let X be a w-solid set of maximal subgroups of G such that ∑ reduces into each element of X. Consider in G the following three subgroups： the ξ-normalizer D of G associated with ∑; the X-prefrattini subgroup W = W（G, X） of G; and a hypercentrally embedded subgroup T of G. Then the lattice ζ（T, W, D） generated by T, D and W is a distributive lattice of pairwise permutable subgroups of G with the cover and avoidance property. This result remains true for the lattice ,ζ（V, W, D）, where V is a subgroup of G whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups of G such that ∑ reduces into V.In this paper, we prove the following result. Let ξ be a saturated formation and ∑ a Hall system of a soluble group G. Let X be a w-solid set of maximal subgroups of G such that ∑ reduces into each element of X. Consider in G the following three subgroups： the ξ-normalizer D of G associated with ∑; the X-prefrattini subgroup W = W（G, X） of G; and a hypercentrally embedded subgroup T of G. Then the lattice ζ（T, W, D） generated by T, D and W is a distributive lattice of pairwise permutable subgroups of G with the cover and avoidance property. This result remains true for the lattice ,ζ（V, W, D）, where V is a subgroup of G whose Sylow subgroups are also Sylow subgroups of hypercentrally embedded subgroups of G such that ∑ reduces into V.

关 键 词：lattice properties PERMUTABILITY FACTORIZATIONS cover and avoidance properties 

分 类 号：O152.7[理学—数学]