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

17 pages