一类有限群的自同构群的正规化分解

Normal Decomposition of Automorphism Groups of a Class of Finite Groups

摘要: 针对有限群的半直积的自同构群的结构问题，运用了Bidwell和Curran于2006年引入的“矩阵表示”这种较为直观的研究方法，结合作者得到的关于稳定自同构群的矩阵公式，以及半直积的自同构群的中心化分解，得到了此类半直积的自同构群具有正规化分解的充分条件，举例说明了条件的不可缺少性，简化了Bidwell关于可裂亚循环p-群的自同构群的相关结论的复杂计算，并给出了定理的相关应用。

Abstract: On the structure problem of automorphism groups for the semi-direct product of finite groups, some hypotheses sufficient to guarantee that semi-direct products of finite groups possess normal decomposition is found by using the relatively intuitive research method of “matrix representation” introduced by Bidwell and Curran in 2006, combined with the matrix formula of stable automorphism group proved by the author, and the centralization decomposition of automorphism group of semi-direct product. Complicated calculation of Bidwell and Curran’s automorphism group of splitting metacyclic p-groups is reduced, the relevant applications of the theorem are given.