Classification of cyclic groups underlying only smooth skew morphisms

Abstract A skew morphism of a finite group A is a permutation φ of A fixing the identity element and for which there is an integer-valued function π on A such that (ab) = (a) ^ (a) (b) φ (a b) = φ (a) φ π (a) (b) for all a, b A a, b ∈ A. A skew morphism φ of A is smooth if the associated power function π is constant on the orbits of φ, that is, ( (a) ) (a) | | π (φ (a) ) ≡ π (a) (mod | φ |) for all a A a ∈ A. In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if n=2ᵉn₁ n = 2 e n 1, where 0 e 4 0 ≤ e ≤ 4 and n₁ n 1 is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.