蕴含交换极大子群的极大类3-群上的光滑斜态射

Smooth skew morphisms of a kind of maximal class 3-groupss which have abelian maximal subgroups

一个有限群G上的斜态射为G的一个置换φ,满足φ(1)=1且φ(gh)=φ(g)φ^(π(g))(h)对任意g,h∈G均成立,其中π为G到集合{1,2,…,d-1}的一个函数且d为φ的阶。若对任意的g∈G都有π(g)=1,则φ为G的自同构。因此斜态射为群的自同构的推广。若对任意g∈G都有π(φ(g))=π(g),则斜态射φ被称为光滑斜态射。本文研究了一类蕴含交换极大子群的极大类3-群上的光滑斜态射,并给出了其完全分类。

A skew morphismφof a finite group G is a permutation of G fixing the identity of G and satisfying the propertyφ(gh)=φ(g)φ^(π(g))(h)for any g,h∈G,whereπis a function from G to{1,2,…,d-1}for the order d ofφ.If for any g∈G,π(g)=1,thenφis an automorphism of G.Hence a skew morphism is a generalization of an automorphism.Whenπ(φ(g))=π(g)for any g∈G,the skew morphismφis called a smooth skew morphism.In this paper,we classify all smooth skew morphisms of a kind of maximal class 3-groups which have abelian maximal subgroups.