Some problems of Wielandt revisited

JOURNAL OF ALGEBRA

卷:302

Article

Knapp, W

关键词: permutation group;    finite;    primitive;    orbital;    suborbit;    subconstituent;    paired;    regular;    faithful;    subnormal subgroup;    simple group;    semisimple group;    generalized fitting subgroup;    nilpotent;    graph;    automorphism group;    edge-transitive;    sylow normalizer;   

DOI  :  10.1016/j.jalgebra.2005.09.013

来源: Elsevier

【 摘 要 】

Several problems in the theory of finite permutation groups considered before by H. Wielandt are attacked by new and traditional methods. One new method is given by the theorem that a semisimple subgroup A of a group G normalizing a different subgroup B isomorphic to A forces that the centralizer in AB of B is non-trivial, hence B is not the generalized Fitting subgroup of its normalizer. This theorem is applied in proving that the paired subconstituent G(alpha)(Delta') (alpha) of a primitive permutation group G is faithful if the non-trivial subconstituent G(alpha)(Delta(alpha)) is regular. If G(alpha)(Delta(alpha)) is a non-abelian simple group all of whose proper subgroups are solvable then the regularity of G(Delta)(alpha(alpha)) even implies that G(alpha)(Delta(alpha)) is faithful. Also several theorems are obtained for the case that a non-trivial subconstituent G(alpha)(Delta(alpha)) is nilpotent or, more generally, that G alpha beta is subnormal in G alpha for beta is an element of Delta(alpha). (c) 2005 Elsevier Inc. All rights reserved.

【 授权许可】

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