Automorphisms of permutational wreath products
The main result of this paper is the following theorem: Let P be a transitive permutation group on the finite set I, \(| I| >1\), let H be the stabilizer in P of a point in I, and S the number of orbits of H on I. Let R be a finite nonabelian indecomposable group. Let \(G=R Wr (P,I)\) and let A be the group of all automorphisms of G which fix the subgroup \(R^ I\). Then Inn(G)\(\leq A\leq Aut(G)\) and \[ | A/Inn(G)| =| Hom(H,Z(R)| | Out(R)| | N_{Aut(P)}(H)/Inn_ H(P)| | Hom(R,Z(R)|^{s-1}. \] Furthermore, \(A=Aut(G)\) if any of the following is satisfied: (1) \(H=1\) and either \(| I| >2\) or R is not a special dihedral group. (2) \(| I| >2\) and H is contained in some core-free maximal subgroup of P. (3) \(O_ 2(P)=1\). (4) R is not of dihedral type. By using the result certain characteristic subgroups and \(| Out(G)|\) are given in the present paper.
scientific article; zbMATH DE number 4061502