Classification of cyclic groups underlying only smooth skew morphisms

IF 0.6 3区数学 Q3 MATHEMATICS Journal of Algebraic Combinatorics Pub Date : 2024-04-03 DOI:10.1007/s10801-024-01311-4

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Abstract

A skew morphism of a finite group A is a permutation $\varphi $ of A fixing the identity element and for which there is an integer-valued function $\pi $ on A such that $\varphi (ab)=\varphi (a)\varphi ^{\pi (a)}(b)$ for all $a, b \in A$ . A skew morphism $\varphi $ of A is smooth if the associated power function $\pi $ is constant on the orbits of $\varphi $ , that is, $\pi (\varphi (a))\equiv \pi (a)\pmod {|\varphi |}$ for all $a\in A$ . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if $n=2^en_1$ , where $0 \le e \le 4$ and $n_1$ is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.

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仅以光滑偏斜变形为基础的循环群的分类

Abstract 有限群 A 的偏斜变形是 A 的一个固定同元素的置换（permutation $\varphi$），对于这个置换，A 上有一个整数值函数 $\pi$，使得 $\varphi(ab)=\varphi(a)\varphi ^{\pi (a)}(b)$ for all $a, b \in A$ 。如果相关的幂函数 $\pi $ 在 $\varphi $ 的轨道上是常数，即 $\pi (\varphi (a))\equiv \pi (a)\pmod {|\varphi |}$ for all $a\in A$ ，那么 A 的倾斜变形 $\varphi $ 是平稳的。在本文中，我们证明了当且仅当 $n=2^en_1$ ，其中 $0 \le e \le 4$ 和 $n_1$ 是奇数无平方数时，阶数为 n 的循环群的每个倾斜态都是光滑的。此外，还给出了非循环无性系群类似问题的部分解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Algebraic Combinatorics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Algebraic Combinatorics provides a single forum for papers on algebraic combinatorics which, at present, are distributed throughout a number of journals. Within the last decade or so, algebraic combinatorics has evolved into a mature, established and identifiable area of mathematics. Research contributions in the field are increasingly seen to have substantial links with other areas of mathematics. The journal publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems.

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