Confirmation for Wielandt's conjecture

TL;DR

This paper proves a conjecture by Wielandt from 1959, showing that under certain conditions involving Hall subgroups and Sylow theorems, the Sylow π-theorem holds for a finite group.

Contribution

It confirms Wielandt's long-standing conjecture by establishing the Sylow π-theorem under specified subgroup conditions in finite groups.

Findings

Proves Wielandt's conjecture from 1959.

Shows the Sylow π-theorem holds when certain Hall subgroups are direct products.

Establishes conditions for conjugacy of maximal π-subgroups.

Abstract

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