学 术 报 告

报告题目:Classification of quotient-balanced skew morphism of cyclic group with minimal auto index

报告学者: Young Soo Kwon教授

报告者单位Yeungnam University,Korea



摘要: A skew morphism of a finite group $A $ is a permutation $\varphi$ on $A$ fixing the identity element of $A$ and for which there exists an integer-valued function $\pi$ on $A$ such that $\varphi(ab) = \varphi(a)\varphi^{\pi(a)}(b)$ for all $a,b\in A$. If $\pi(a)=1$ for all $a\in A$, then $\varphi$ is an automorphism of $A$. Thus, skew morphisms can be viewed as a generalization of group automorphisms. If $h$ is the smallest positive integer such that  $\varphi^h$  is an automorphism, then we call $h$ {\it auto-index} of $\varphi$, especially when $h$ is 2, we call $\varphi$ a proper s quare root of an  automorphism.  The \textit{period} of $\varphi$ is the smallest positive integer $d$ such that $\varphi^d(a)=\pi(a)$ for all $a \in A$. One can easily show that the period $d$ is a divisor of the auto-index $h$. If thee two numbers are the same, we say that  $\varphi$ has minimal auto-index $h$. For a skew morphism of $A$, it is known that the induced skew morphism $\overline\varphi$  of $\overline A=A/\Core\varphi$ is well defined. If the induced skew orphism $\overline\varphi$ is an automorphism, then we call $\varphi$ {\it quotient-balanced } skew morphism of $A$.

In this talk, we will consider skew morphism of cyclic groups and talk about classification of quotient-balanced skew morphism of cyclic group with minimal auto-index. This is a joint work with Kan Hu and Jun-Yang Zhang.