On cyclic actions of finite groups (Q6623417)

Let \(G\) be a finite group and let \(A\) be a group acting on \(G\). Let \(\mathfrak{F}_{p}\) be the class of finite groups that are extensions of a \(p\)-group via a group of exponent that divides \(p-1\) where \(p\) is a prime. The \(\mathfrak{F}_{p}\)-residual \(A^{\mathfrak{F}_{p}}\) is the intersection of all normal subgroups \(N\) of \(A\) with \(A/N \in \mathfrak{F}_{p}\).\N\NThe group \(A\) is said to act \(p\)-cyclically on \(G\) if \(A^{\mathfrak{F}_{p}}\) acts stably on \(G/O_{p'}(G)\), and \(A\) is said to act cyclically on \(G\) if \(A\) acts \(p\)-cyclically on \(G\) for all primes \(p\). In the paper under review, the actions of a group \(A\) on a \(p\)-group or a \(p\)-soluble group \(G\) are investigated and some criteria of an action to be cyclic or \(p\)-cyclic are obtained.