【摘要】：在有限群的研究中,利用群的階數(shù),子群和元素的性質(zhì)等方面來(lái)刻畫群的組合問(wèn)題,一直以來(lái)都是研究有限群論的一個(gè)重要方向.在關(guān)于有限群的組合問(wèn)題中,研究群的因子分解是一件有趣和有意義的事情,并且該方面的研究與有限群的概率問(wèn)題有密切地聯(lián)系.特別地,有限群的子群交換度可以借助于有限群的因子分解數(shù)來(lái)確定,反之亦然.設(shè)G是一個(gè)有限群,A和B是G的兩個(gè)子群,若G=AB,則稱G被A和B因子分解.本文主要通過(guò)群的階,元素的階,莫比烏斯反演公式,Abel p-群的子群數(shù)來(lái)研究有限p-群的因子分解.本文主要包括三個(gè)部分,第一章是介紹有限群的因子分解和有限群子群的交換度的研究背景及意義和一些預(yù)備知識(shí).第二章計(jì)算一類廣義超特殊p-群,即(?),m≥1,的因子分解數(shù).在第三章,計(jì)算一類內(nèi)交換p-群,即(?),n＞m≥1的因子分解數(shù).所用的方法是:先根據(jù)子群A,B是否包含導(dǎo)群G'進(jìn)行分析、計(jì)算、討論,這里分為三種情況,第一種情況,子群A和B都包含導(dǎo)群G';第二種情況,子群A和B都不包含導(dǎo)群G';第三種情況,子群A包含導(dǎo)群G',子群B不包含導(dǎo)群G',反之亦然.然后分別計(jì)算出這三種情況中滿足條件的G的因子分解數(shù),最后求得G的因子分解數(shù).
[Abstract]:In the study of finite groups, it has always been an important direction to study the combinatorial problem of finite groups by using the order, subgroups and properties of elements to characterize the combinatorial problems of groups. In the combinatorial problem of finite groups, it is interesting and meaningful to study the factorization of groups, and the research in this field is closely related to the probability problem of finite groups. In particular, the degree of subgroup commutation of a finite group can be determined by the factorization number of a finite group, and vice versa. Let G be a finite group and A and B be two subgroups of G. If G is a finite group, then G is decomposed by A and B factors. In this paper, the factorization of finite p-groups is studied by means of the order of groups, the order of elements, the Mobius inversion formula, and the number of subgroups of Abel p-groups. This paper mainly consists of three parts. The first chapter introduces the research background and significance of the factorization of finite groups and the commutation degree of finite group subgroups and some preparatory knowledge. In chapter 2, we calculate the factorization numbers of a class of generalized super-special p-groups, that is, (?), m 鈮，

本文編號(hào)：2446948