Theorems like Wielandt's
A well-known theorem of Wielandt asserts that a 3/2-transitive permutation group of a finite set X acts primitively or as a Frobenius group on X. This result reduces the structure of a transitive permutation group G of a finite set X, provided that the two-point stabilizers of G all have equal order. Does there exist a similar result if G does not necessarily act transitively on X? A partial answer is given in the second theorem of the present paper. If, in addition, the two-point stabilizers of G are conjugate in G, the author gets a complete classification, provided that \(F(G)=1.\) Theorem 1. Let G be a permutation group of a finite set X. Assume that the two-point stabilizers of G are non-trivial and conjugate in G. If \(F(G)=1\), then G acts doubly transitively on X or there exists \(e\in {\mathbb{N}}\setminus \{0,1,2\}\) such that \(F^*(G)\cong PSL(2,2^ e).\) Theorem 2. Let G be a permutation group of a finite set X. Assume that the two-point stabilizers of G all have equal order \(>1\). If all two- point stabilizers of G are nilpotent, then G acts 3/2-transitively on X.
scientific article; zbMATH DE number 4191652