一种传递群的构造举例

Construction of an Example of a Transitive Group

摘要: 对于传递置换群G在对应的对称群中的正规化子N，N依共轭作用在子群G上，得到一个同态Ψ:N→Aut(G)。Dixon在经典专著《Permutation Groups》中对此同态的同态像与G的自同构群的关系进行了研究，描述了位于ImΨ下G的全体自同构。在此基础上Dixon提出构造一个例子，使得对于传递群G，满足Ψ的像不是G的全体自同构。本文我们在四次对称群上提供这样一个例子。

Abstract: In this paper, we will consider the normalizer of the subgroup of the transitive permutation groups. The N of G acts naturally on the set G by conjugation, this gives a homomorphism Ψ:N→Aut(G). Dixon studied the relationship between the homomorphic image and the automorphisms of G in his monograph Permutation Groups. He described the automorphisms of G which lie in the image of Ψ. Based on this, he proposed to construct an example G for which the image of this homomorphism is not all of automorphisms of G. In this short note, we will provide such an example on the quartic symmetry group.