Cyclic complementary extensions and skew-morphisms

A cyclic complementary extension of a finite group $A$ is a finite group $G$ which contains $A$ and a cyclic subgroup $C$ such that $A\cap C=\{1_G\}$ and $G=AC$. For any fixed generator $c$ of the cyclic factor $C=\langle c\rangle$ of order $n$ in a cyclic complementary extension $G=AC$, the equations $cx=\varphi(x)c^{\Pi(x)}$, $x\in A$, determine a permutation $\varphi:A\to A$ and a function $\Pi:A\to\mathbb{Z}_n$ on $A$ characterized by the properties: (a) $\varphi(1_A)=1_A$ and $\Pi(1_A)\equiv1\pmod{n}$; (b) $\varphi(xy)=\varphi(x)\varphi^{\Pi(x)}(y)$ and $\Pi(xy)\equiv\sum_{i=1}^{\Pi(x)}\Pi(\varphi^{i-1}(y))\pmod{n}$, for all $x,y\in A$. The permutation $\varphi$ is called a skew-morphism of $A$ and has already been extensively studied. One of the main contributions of the present paper is the recognition of the importance of the function $\Pi$, which we call the extended power function associated with $\varphi$. We show that {\em every} cyclic complementary extension of $A$ is determined and can be constructed from a skew-morphism $\varphi$ of $A$ and an extended power function $\Pi$ associated with $\varphi$. As an application, we present a classification of cyclic complementary extensions of cyclic groups obtained using skew-morphisms which are group automorphisms.