Cyclic complementary extensions and skew-morphism

Abstract A cyclic complementary extension of a finite group 𝐴 is a finite group 𝐺 which contains 𝐴 and a cyclic subgroup 𝐶 such that A ∩ C = { 1 G } A\cap C=\{1_{G}\} and G = A ⁢ C G=AC . For any fixed generator 𝑐 of the cyclic factor C = ⟨ c ⟩ C=\langle c\rangle of order 𝑛 in a cyclic complementary extension G = A ⁢ C G=AC , the equations c ⁢ x = φ ⁢ ( x ) ⁢ c Π ⁢ ( x ) cx=\varphi(x)c^{\Pi(x)} , x ∈ A x\in A , determine a permutation φ : A → A \varphi\colon A\to A and a function Π : A → Z n \Pi\colon A\to\mathbb{Z}_{n} on 𝐴 characterized by the following properties: φ ⁢ ( 1 A )