The strong \(\pi \)-Sylow theorem for the groups \(\operatorname{PSL}_2(q) \) (Q6631346)

Let \(G\) be a finite group, \(\pi\) a set of primes and \(\pi(G)\) the set of primes dividing \(|G|\), the order of \(G\). The group \(G\) is a \(\pi\)-group if \(\pi(G) \subseteq \pi\). Following \textit{H. Wielandt} [in: Proceedings of the International Congress of Mathematicians, Edinburgh, 14--21 August 1958. Cambridge: At the University Press. 268--278 (1960; Zbl 0122.27303)], the \(\pi\)-Sylow theorem holds for \(G\) if all maximal \(\pi\)-subgroups of \(G\) are conjugate and if the \(\pi\)-Sylow theorem holds for every subgroup of \(G\), then the strong \(\pi\)-Sylow theorem holds for \(G\). A question (raised by \textit{H. Wielandt} [Proc. Symp. Pure Math. 37, 161--173 (1980; Zbl 0458.20024)]) then naturally arises. Which finite simple non-abelian groups satisfy the conclusions of the strong \(\pi\)-Sylow theorem? By now the answer is known for sporadic and alternating groups (see \textit{N. Ch. Manzaeva} [Sib. Èlektron. Mat. Izv. 9, 294--305 (2012; Zbl 1329.20019)] and the first author [Algebra Logic 47, No. 3, 210--227 (2008; Zbl 1155.20018)]).\N\NIn the paper under review, the author gives some arithmetic criteria for the validity of the strong \(\pi\)-Sylow theorem for the groups \(\mathrm{PSL}_{2}(q)\). The authors remark that, excluding the Suzuki groups \(\mathrm{ Sz}(2^{2n+1})\), each group \(L\) of Lie type with base field of \(q\) elements has a section isomorphic to \(\mathrm{PSL}_{2}(q)\). So the conditions they determine are necessary for the \(\pi\)-Sylow theorem to hold for \(L\).