【作者】 陈瑞芳；

【导师】 郭秀云；

【作者基本信息】 上海大学 ， 基础数学， 2014， 博士

【摘要】 有限群的共轭类长以及共轭类的个数都与有限群的结构有着非常紧密的联系,众多群论工作者都参与到这一领域的研究,获得了许多重要的研究成果.近年来,人们在如何应用有限群G的真正规子群所包含的G-共轭类的个数来确定有限群G的结构方面进行了许多有益的尝试.本学位论文中的一部分就是研究有限群G的全部非平凡正规子群所包含的G-共轭类个数恰为三个连续整数时G的结构.实际上,在第三章,我们研究了有限群G的全部真正规子群所包含的G-共轭类个数的集合恰为{1,2,3,4}的情形,并给出了满足此条件的有限非完备群的完全分类.进一步,我们也研究了有限群G的全部真正规子群所包含的G-共轭类个数的集合恰为{1,m,m+1,m+2}的情形,并得到了满足此条件的有限非完备群的结构(这里m是任一大于2的正整数).本学位论文的另一部分就是希望利用有限群中部分元素的共轭类长来研究有限群的结构.设N是有限群G的一个正规子群.我们主要是借助N中某些元素的G-共轭类长研究N的结构.实际上,在第四章,通过借助有限群的共轭类图的有关性质,我们研究了p-可解正规子群N的p’-元的G-共轭类长与N的p-补的结构的关系.在第五章,我们研究了π-可分群G的正规子群N的素数或双素数π’-元的G-共轭类长对N的π-补的结构和性质的影响.更多还原

【Abstract】 The conjugacy class sizes and conjugacy class numbers of a finite group have close relationship with the structure of a finite group, and many group workers have participated in this area, and lots of important results have been obtained. In recent years, many workers try to determine the structure of a finite group G by using the numbers of G-conjugacy classes contained in its proper normal subgroups. One part of this thesis is to investigate the structure of a finite group G when the numbers of G-conjugacy classes contained in its all non-trivial normal subgroups are three consecutive integers. In fact, in Chapter3, we investigate the structure of a finite group G when the set of numbers of G-conjugacy classes contained in its all proper normal subgroups is{1,2,3,4}, and we give a complete classification of finite non-perfect groups which satisfy this condition. Furthermore, we also investigate the structure of a finite group G when the set of numbers of G-conjugacy classes contained in its all proper normal subgroups is{1, m, m+1, m+2}, and we obtain the structure of a finite non-perfect group in this case (where m is an arbitrary integer larger than2).Another part of this thesis is to investigate the structure of a finite group G by using the conjugacy class sizes of part of its elements. Suppose that N is a normal subgroup of G. We mainly investigate the structure of N by using the G-conjugacy class sizes of some elements in N. In fact, in Chapter4, by using the properties of conjugacy class graph of a finite group, we investigate the relationship between the G-conjugacy class sizes of p’-elements in a p-solvable normal subgroup N and the structure of p-complements of N. In Chapter5, we investigate the influence of the G-conjugacy class sizes of primary and biprimary7r’-elements in a normal subgroup N of a7r-separable group G on the structure and properties of7r-complements of N.更多还原