循环$2$-群上斜型四的斜同态的分类与枚举
Classification and enumeration of skew morphisms of skew-type four on cyclic $2$-groups
浏览论文内容
中文总结 AI 辅助
本文构造、分类并枚举循环$2$-群上斜型四的斜同态，给出所有此类斜同态的显式公式及数量闭式表达式。
中文摘要 AI 辅助
有限群$A$上的斜同态是$A$上的一个置换$\varphi$，它固定$A$的单位元，并且存在一个整值函数$\pi:A\to\mathbb{Z}_{|\varphi|}$，使得对所有$x,y\in A$，有$\varphi(xy)=\varphi(x)\varphi^{\pi(x)}(y)$。$\varphi$的核是子群$\Ker\varphi=\{x\in A\mid \pi(x)=1\}$，指数$[A:\Ker\varphi]$称为$\varphi$的斜型。本文构造、分类并枚举了循环$2$-群上斜型四的斜同态。我们的主要结果给出了所有此类斜同态的显式公式及其数量的闭式表达式。
英文摘要
A skew morphism on a finite group $A$ is a permutation $φ$ on $A$ that fixes the identity element of $A$ and for which there exists an integer-valued function $π:A\to\mathbb{Z}_{|φ|}$ such that $φ(xy)=φ(x)φ^{π(x)}(y)$ for all $x,y\in A$. The kernel of $φ$ is the subgroup $\Kerφ=\{x\in A\mid π(x)=1\}$, and the index $[A:\Kerφ]$ is called the skew-type of $φ$. In this paper we construct, classify and enumerate the skew morphisms of skew-type four on cyclic $2$-groups. Our main results give explicit formulas for all such skew morphisms and closed-form expressions for their numbers.
发表机构
- Zhejiang Ocean University（浙江海洋大学）
机构由 AI 辅助整理，请以论文原文为准。
