On a theorem of P. Hall. (Q2856529)

Let \(A\) be a finite group acting on a finite group \(G\) and let \(AG\) denote the semidirect product. A mapping \(\varphi\) from \(A\) to \(G\) is said to be a crossed homomorphism with respect to the action of \(A\) on \(G\) when \(\varphi(ab)=\varphi(a)^b\varphi(b)\) for all elements \(a,b\in A\).NEWLINENEWLINE The authors are inspired by a theorem of Philip Hall to prove, by means of character theory, that the number of crossed homomorphisms from \(A\) to \(G\) is a multiple of \(\gcd(|A/B|,|G|)\), provided that \(B\) is a normal subgroup of \(A\) such that \(A/B\) is cyclic. In the particular case in which \(A\) acts trivially on \(G\), the authors recover a classic Frobenius' result which asserts that for any finite group \(G\) the number of \(n\)-th roots of unity is a multiple of \(\gcd(n,|G|)\). Some other corollaries are obtained, and among them, a new proof of a result by \textit{P. J. Cameron} and \textit{T. W. Müller} [Arch. Math. 82, No. 3, 200-204 (2004; Zbl 1072.20060)] is provided.