On the number of invariant Hall subgroups under coprime action (Q1702158)

Let \(A\) be a finite group that acts coprimely on a finite \(\pi\)-separable group \(G\) via automorphisms. The author obtains a divisibility property concerning the number of \(A\)-invariant Hall \(\pi\)-subgroups. Precisely, he proves that if \(H\) is an \(A\)-invariant subgroup of \(G\) and \(V\) and \(U\) are \(A\)-invariant \(\pi\)-subgroups of \(G\) such that \(V\leq U\leq H\), then the number of \(A\)-invariant Hall \(\pi\)-subgroups of \(H\) containing \(U\) divides the number of \(A\)-invariant Hall \(\pi\)-subgroups of \(G\) containing \(V\). This result generalizes and simplifies some results achieved previously. The divisibility property for the particular case \(A=1\) and \(V=U\) was obtained in [\textit{M. J. Iranzo} et al., Commun. Algebra 33, No. 8, 2713--2716 (2005; Zbl 1077.20036)]. On the other hand, the reviewer and \textit{C. Shao} addressed the same problem under the coprime action setting for a single prime, \(\pi=\{p\}\), in a solvable group [J. Algebra 490, 380--389 (2017; Zbl 1385.20010)]. The root of these investigations lays in a theorem of G. Navarro that asserts that if \(G\) is a \(p\)-solvable group and \(H\) is a subgroup of \(G\), then the number of Sylow \(p\)-subgroups of \(H\) divides the number of Sylow \(p\)-subgroups of \(G\).