Some problems of Wielandt revisited

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 of a primitive permutation group G is faithful if the non-trivial subconstituent is regular. If is a non-abelian simple group all of whose proper subgroups are solvable then the regularity of even implies that is faithful. Also several theorems are obtained for the case that a non-trivial subconstituent is nilpotent or, more generally, that Gαβ is subnormal in Gα for β∈Δ(α).