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On the number of invariant Hall subgroups under coprime action. (English) Zbl 1388.20034

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 [M. J. Iranzo et al., Commun. Algebra 33, No. 8, 2713–2716 (2005; Zbl 1077.20036)]. On the other hand, the reviewer and 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 \(G\).

MSC:

20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:

[1] A. Beltrán and C. Shao, On the number of invariant Sylow subgroups under coprime action, J. Algebra 490 (2017), 380-389. · Zbl 1385.20010 · doi:10.1016/j.jalgebra.2017.07.005
[2] I. M. Isaacs, Finite Group Theory, Amer. Math. Soc., Providence, RI, 2008. · Zbl 1169.20001 · doi:10.1090/gsm/092
[3] M. Iranzo, F. Monasor, and J. Medina, Arithmetical questions in \[\pi\] π-separable groups, Comm. Algebra 33 (2005), 2713-2716. · Zbl 1077.20036 · doi:10.1081/AGB-200063998
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