On the number of invariant Sylow subgroups under coprime action (Q2401719)

In this paper and in this review, only finite groups are considered. The general assumption is that a group \(A\) acts on a group \(G\) and that \(\gcd(|A|, |G|)=1\). Let \(\nu_p^A(G)\) denote the number of \(A\)-invariant Sylow \(p\)-subgroups of \(G\). The aim of this paper is to obtain some arithmetical properties on these numbers. The first main result of the paper, Theorem~A, can be regarded as a generalisation of a result of [\textit{M. Hall}, J. Algebra 7, No. 3, 363--371 (1967; Zbl 0178.02102)]: If \(N\) is an \(A\)-invariant normal subgroup of \(G\), then for every prime \(p\) and every \(A\)-invariant Sylow \(p\)-subgroup \(P\) of \(G\) we have that \(\nu_p^A(G)=\nu_p^A(G/N)\nu_p^A(N)\nu_p^A(\text{N}_{PN}(P\cap N))\). In Theorem~B, it is shown that if \(H\) is an \(A\)-invariant subgroup of \(G\) and \(\text{C}_G(A)\leq \text{N}_G(H)\), then \(\nu_p^A(H)\) divides \(\nu_p^A(G)\) for every prime \(p\). In general, for every \(A\)-invariant subgroup \(H\) of \(G\), it is shown that \(\nu_p^A(H)\leq \nu_p^A(G)\) for every prime \(p\). For soluble groups, this result is improved in Theorem~C: If \(G\) is a soluble finite group acted on comprimely by a finite group \(A\) and \(H\) is an \(A\)-invariant subgroup of \(G\), then \(\nu_p^A(H)\) divides \(\nu_p^A(G)\) for every prime \(p\). The solubility hypothesis cannot be removed, as shown in one example. The authors present some examples showing that some arithmetical properties that one could expect do not hold. For instance, \(\nu_p^A(G)\) need not be congruent to~\(1\) modulo~\(p\). Lemma~2.2 shows that if the action of \(A\) on~\(G\) is coprime, then \(\nu_p^A(G)\) divides \(\nu_p(G)\), but this property does not hold when the action is not coprime.