Classification of cyclic groups underlying only smooth skew morphisms (Q6570590)

Let \(A\) be a finite group. A skew morphism of \(A\) is a permutation \(\varphi\) of \(A\) such that \(\varphi(1)=1\) and for which there is a function \(\pi: A \rightarrow \mathbb{Z}\) such that \(\varphi(ab) =\varphi(a)\varphi^{\pi(a)}(b)\) for all \(a,b \in A\). A skew morphism \(\varphi\) of \(A\) is smooth if \(\pi(\varphi(a)) \equiv \pi(a) \mod |\varphi|\) for all \(a \in A\) (that is the associated power function \(\pi\) is constant on the orbits of \(\varphi\)).\N\NThe main result in the paper under review is that every skew morphism of a cyclic group of order \(n\) is smooth if and only if \(n=2^{e}n_{1}\), where \(0 \leq e \leq 4\) and \(n_{1}\) is an odd square-free number.\N\NFor non-cyclic abelian groups, the authors obtain the following partial result (Theorem 1.2): Let \(A\) be a non-cyclic abelian group of order \(n=2^{f}n_{1}\), where \(f \geq 0\) and \(n_{1}\) is odd. If \(A\) underlies only smooth skew morphisms, then \(n_{1}\) is square-free and the Sylow \(2\)-subgroup of \(A\) contains no direct factors isomorphic to \(\mathbb{Z}_{2^{e}}\) (\(e \geq 5\)).