Classification of cyclic groups underlying only smooth skew morphisms

Abstract

AbstractA skew morphism of a finite group A is a permutation $$\varphi $$ φ of A fixing the identity element and for which there is an integer-valued function $$\pi $$ π on A such that $$\varphi (ab)=\varphi (a)\varphi ^{\pi (a)}(b)$$ φ ( a b ) = φ ( a ) φ π ( a ) ( b ) for all $$a, b \in A$$ a , b ∈ A . A skew morphism $$\varphi $$ φ of A is smooth if the associated power function $$\pi $$ π is constant on the orbits of $$\varphi $$ φ , that is, $$\pi (\varphi (a))\equiv \pi (a)\pmod {|\varphi |}$$ π ( φ ( a ) ) ≡ π ( a ) ( mod | φ | ) for all $$a\in A$$ a ∈ A . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if $$n=2^en_1$$ n = 2 e n 1 , where $$0 \le e \le 4$$ 0 ≤ e ≤ 4 and $$n_1$$ n 1 is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.