论文标题
循环群的偏斜形态的商
Quotients of skew morphisms of cyclic groups
论文作者
论文摘要
有限的组$ b $的偏斜形态是$ b $的排列$φ$ $ b $，它保留了$ b $的身份元素，并且拥有一个$ a \ in b $中的每个$ a \ in B $中存在一个正整数$ i_a $，因此$φ（ab）=φ（a）=φ（a）φ^{i_a}（i_a}（i_a}（i_a}（i_a}）众所周知，对所有有限循环群的偏斜形态进行分类的问题非常困难，没有此类分类最新。 $ \ mathbb {z} _n $的每个偏斜的形态$φ$ $φ$与$ \ mathbb {z} _ {| \！\！\langleφ\ rangle \！|} $的特定偏斜形态密切相关。在本文中，我们使用这种关系和其他观察结果来证明有关有限循环基团偏斜形态的新定理。特别是，我们将所有循环组的偏差分类为$ 2^em $的所有循环群，并用$ e \ in \ {0,1,1,2,3,4 \} $和$ m $ odd and Square-free。我们还开发了一种用于查找环状群体偏斜形态的算法，并在岩浆中实施该算法，以获得最高$ 161 $的循环循环群的所有偏斜形态的普查。在制备本文期间，我们注意到纸循环补充的第5节中存在一些缺陷，并且偏向组的形态[J.代数453（2016），68-100]。我们提出并证明了有问题的原始断言的较弱版本（即引理5.3（b），定理5.6和推论5.7），并表明我们的修改可用于修复所有结果的证明（在上述纸张中），这些证明使用至少一个有问题的断言。
A skew morphism of a finite group $B$ is a permutation $φ$ of $B$ that preserves the identity element of $B$ and has the property that for every $a\in B$ there exists a positive integer $i_a$ such that $φ(ab) = φ(a)φ^{i_a}(b)$ for all $b\in B$. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism $φ$ of $\mathbb{Z}_n$ is closely related to a specific skew morphism of $\mathbb{Z}_{|\!\langle φ\rangle\!|}$, called the quotient of $φ$. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order $2^em$ with $e\in \{0,1,2,3,4\}$ and $m$ odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to $161$. During the preparation of this paper we noticed a few flaws in Section~5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68-100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.
