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 } and G = A ⁢ C . For any fixed generator 𝑐 of the cyclic factor C = ⟨ c ⟩ of order 𝑛 in a cyclic complementary extension G = A ⁢ C , the equations c ⁢ x = φ ⁢ ( x ) ⁢ c Π ⁢ ( x ) , x ∈ A , determine a permutation φ : A → A and a function Π : A → Z n on 𝐴 characterized by the following properties:

1. φ ⁢ ( 1 A ) = 1 A and Π ⁢ ( 1 A ) ≡ 1 ⁢ ( mod ⁢ n ) ;

2. φ ⁢ ( x ⁢ y ) = φ ⁢ ( x ) ⁢ φ Π ⁢ ( x ) ⁢ ( y ) and Π ⁢ ( x ⁢ y ) ≡ ∑ i = 1 Π ⁢ ( x ) Π ⁢ ( φ i − 1 ⁢ ( y ) ) ⁢ ( mod ⁢ n ) for all x , y ∈ 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 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.