Cyclic complementary extensions and skew-morphisms

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 $A\cap C=\{1_G\}$ 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 $cx=φ(x)c^{Π(x)}$, $x\in A$, determine a permutation $φ:A\to A$ and a function $Π:A\to\mathbb{Z}_n$ on $A$ characterized by the properties: (a) $φ(1_A)=1_A$ and $Π(1_A)\equiv1\pmod{n}$; (b) $φ(xy)=φ(x)φ^{Π(x)}(y)$ and $Π(xy)\equiv\sum_{i=1}^{Π(x)}Π(φ^{i-1}(y))\pmod{n}$, for all $x,y\in A$. The permutation $φ$ 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 $Π$, which we call the extended power function associated with $φ$. We show that {\em every} cyclic complementary extension of $A$ is determined and can be constructed from a skew-morphism $φ$ of $A$ and an extended power function $Π$ associated with $φ$. As an application, we present a classification of cyclic complementary extensions of cyclic groups obtained using skew-morphisms which are group automorphisms.