Some problems of Wielandt revisited. (Q852667)

Helmut Wielandt's notebooks contain results, mostly about group theory, which are not published or only partially available. There is a project under way to change this and to make his work easily accessible. While working on this project, the author achieved improvements on several of Wielandt's theorems and on his own past results. The main tools are a theorem of Wielandt about paired subconstituents of a finite primitive permutation group and the new result stating that for \(A\) and \(B\) isomorphic subgroups of a finite group \(G\), \(A\) a direct product of non-Abelian simple groups, \(A\) normalizing \(B\) and \(A\not\subset B\), we have that the centralizer in \(AB\) of \(B\) is not trivial. The author for instance proves that, for a primitive permutation group \(G\), if the paired subconstituent \(G_\alpha^{\Delta(\alpha)}\) is regular, then \(G_\alpha^{\Delta'(\alpha)}\) is faithful; moreover if \(G_\alpha^{\Delta(\alpha)}\) is a non-Abelian simple group all of whose proper subgroups are solvable, then \(G_\alpha^{\Delta'(\alpha)}\) is also faithful. This improves on a long-standing conjecture based on work by Rietz and Weiss, on which Wielandt also worked.