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Cyclic complementary extensions and skew-morphism
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DOI：10.1515/jgth-2024-0144.png)
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Abstract
En 中文
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 boolean AND 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 = phi ( x ) c Pi ( x ) cx=\varphi(x)c{\Pi(x)} , x is an element of A x\in A , determine a permutation phi : A -> A \varphi\colon A\to A and a function Pi : A -> Z n \Pi\colon A\to\mathbb{Z}_{n} on A characterized by the following properties:phi ( 1 A ) = 1 A \varphi(1_{A})=1_{A} and Pi ( 1 A ) equivalent to 1 ( mod n ) \Pi(1_{A})\equiv 1\ (\mathrm{mod}\ n) ; phi ( x y ) = phi ( x ) phi Pi ( x ) ( y ) \varphi(xy)=\varphi(x)\varphi{\Pi(x)}(y) and Pi ( x y ) equivalent to & sum; i = 1 Pi ( x ) Pi ( phi i - 1 ( y ) ) ( mod n ) \Pi(xy)\equiv\sum_{i=1}{\Pi(x)}\Pi(\varphi{i-1}(y))\ (\mathrm{mod}\ n) for all x , y is an element of A 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 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.
Keywords:
REGULAR CAYLEY MAPS
FACTORIZATIONS
CLASSIFICATION
Journal
J
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0.5
Papers:
38
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0