关于内幂零群结构定理的一个注记  

A Note on the Structural Theorem of Minimal Nonnilpotent Groups

作　　者：王玉婷[1] 郝成功[1] WANG Yu-ting HAO Cheng-gong(School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Chin)

机构地区：[1]山西大学数学科学学院,山西太原030006

出　　处：《中北大学学报（自然科学版）》2017年第2期99-102,共4页Journal of North University of China(Natural Science Edition)

基　　金：山西省自然科学基金资助项目(201601D011006)

摘　　要：研究了极小非平凡的群作用.将域F上有限维向量空间线性变换不可约的等价条件推广到初等交换p-群上,再结合极小非平凡作用的定义,得到了Hall-Higman简化定理的充要条件形式,从而给出了极小非平凡作用的另一种刻画,利用此种刻画探讨了p-群的一个p′-自同构何时在Frattini商群上的诱导作用不可约,重新证明了Schmidt定理.作为上述两个结果的综合应用,给出了内幂零群结构定理的一个新的描述和简化证明.Minimal nontrivial actions were studied. The equivalent conditions of irreducibility of linear transformations of finite dimensional vector spaces over a field F were generalized to the elementary abe- lian p-groups. Combining with the concept of minimal nontrivial actions, a necessary and sufficient con- dition of Hall-Higman＇s theorem was obtained. As a corollary, a new characterization of the minimal nontrivial actions was given. This new characterization was applied to study when the induced action of a p＇-automorphism of a p-group on the Frattini quotient is irreducible. Furthermore, a new proof of Schmidt＇s theorem is obtained. As a consequence, a new criterion （with a simplified proof） of minimal nonnilpotent groups is developed.

关 键 词：内幂零群 极小非平凡作用 不可约 自同构 

分 类 号：O152.1[理学—数学]