Confirmation for Wielandt's conjecture. (Q2343513)

A finite group \(G\) is said to satisfy \(\mathcal D_\pi\) (or is a \(\mathcal D_\pi\)-group) when \(G\) contains a Hall \(\pi\)-subgroup and all maximal \(\pi\)-subgroups of \(G\) are conjugate. This property is equivalent to satisfying the complete analog of Sylow's theorem for Hall \(\pi\)-subgroups of a group. A classical result of Wielandt's asserts that if a group \(G\) possesses a nilpotent Hall \(\pi\)-subgroup for a set of primes \(\pi\), then \(G\) satisfies \(\mathcal D_\pi\). There exist many generalizations and analogs of Wielandt's theorem and one of the earliest was obtained by Wielandt himself and claims as follows: Suppose that \(\pi\) is a union of disjoint subsets \(\sigma\) and \(\tau\) and assume that a group \(G\) possesses a Hall \(\pi\)-subgroup \(H=H_\sigma\times H_\tau\), where \(H_\sigma\) is a nilpotent \(\sigma\)-subgroup and \(H_\tau\) is a \(\tau\)-subgroup of \(H\), and let \(G\) satisfy \(\mathcal D_\tau\). Then \(G\) satisfies \(\mathcal D_\pi\). Wielandt asked whether, instead of the nilpotency of \(H_\sigma\), it would be enough to assume that \(G\) satisfies \(\mathcal D_\pi\). The main result of the paper under review is the following, and in particular, completely confirms Wielandt's conjecture. Let a set \(\pi\) of primes be a union of disjoint subsets \(\sigma\) and \(\tau\). Assume that a finite group \(G\) possesses a Hall \(\pi\)-subgroup \(H=H_\sigma\times H_\tau\), where \(H_\sigma\) and \(H_\tau\) are \(\sigma\)- and \(\tau\)-subgroups, respectively. Then \(G\) satisfies \(\mathcal D_\pi\) if and only if \(G\) satisfies both \(\mathcal D_\sigma\) and \(\mathcal D_\tau\). The proof of this theorem reduces to simple groups and it strongly depends on the criterion for a simple group to satisfy \(\mathcal D_\pi\), which was previously obtained by the second author [\textit{D. O. Revin}, Algebra Logika 47, No. 3, 364-394 (2008); translation in Algebra Logic 47, No. 3, 210-227 (2008; Zbl 1155.20018)].