Confirmation for Wielandt's conjecture

Abstract

Let $π$ be a set of primes. By H.Wielandt definition, {\it Sylow $π$-theorem} holds for a finite group $G$ if all maximal $π$-subgroups of $G$ are conjugate. In the paper, the following statement is proven. Assume that $π$ is a union of disjoint subsets $σ$ and $τ$ and a finite group $G$ possesses a $π$-Hall subgroup which is a direct product of a $σ$-subgroup and a $τ$-subgroup. Furthermore, assume that both the Sylow $σ$-theorem and $τ$-theorem hold for $G$. Then the Sylow $π$-theorem holds for $G$. This result confirms a conjecture posed by H.\,Wielandt in~1959.