中心非循环的自同构群的若干研究
发布时间：2018-12-09 10:04
【摘要】：在群论的研究领域中,有限p-群的自同构群阶的最佳下界一直是一个热点问题,关于最佳下界有一个著名的LA-猜想,即设G是有限非循环p-群,|G| =pn,n2,则一定有|G|||Aut(G)|.满足LA-猜想的群称为LA-群.本文立足于Rodney James的p~6阶群的分类理论基础上,进一步展开对LA-猜想的研究工作.本文拟给出了一系列由p~6阶群推广的中心商群同构于第16家族和第17家族但中心非循环的有限p-群,由此验证扩张群是否为LA-群.具体方法如下:首先,根据p-群和中心商群的结构,得出一些同构于第16家族和第17家族群满足的关系式;其次,判断该群的存在性,通过反证法排除不存在的群,存在的群则利用Schreier群扩张理论和Van Dyek自由群理论证明其存在性;最后,讨论扩张后的新群的自同构群的下界,即验证 LA-猜想.为证|G|||Aut(G)|,选取Aut(G)的一个子群R(G)= Ac(G)Inn(G),从而转化成论证|G|||R(G)|,进而得到|G|||Aut(G)|,最终得到若干中心非循环且中心商群的阶为p~6的有限非循环p-群是LA-群.即在Φ_(16)到Φ_(17)这两个家族的群中,找出存在中心非循环且中心商群的阶为p~6的LA-群G,使得G/Z(G)(?)H,其中H∈Φ_(16)-Φ_(17).本文主要结果:(1)当H=Φ_(16)(16),Φ_(16)(2211)b,Φ_(16)(2211)fr时,存在中心非循环的有限p-群G,使得中心商群G/Z(G)(?)H,并且G是LA-群;(2)当H=Φ_(17)(16),Φ_(17)(2211)f,Φ_(17)(2211)mr,mr,s时,及p =3,H=Φ_(17)(214)C,Φ_(17)(214)br,Φ_(17)(214)d时,存在中心非循环的有限p-群G,使得中心商群G/Z(G)(?)H,并且G是LA-群.
[Abstract]:In the research field of group theory, the optimal lower bound of automorphism group of finite p-group has always been a hot issue. There is a famous LA- conjecture about the best lower bound, that is, let G be a finite aperiodic p-group, and G = pn,n2,. Then there must be a G Aut (G). A group satisfying LA- 's conjecture is called a LA- group. Based on the classification theory of Rodney James's group of order 6, this paper further studies the conjecture of LA-. In this paper, we give a series of finite p- groups whose central quotient groups, generalized by groups of order 6, are isomorphic to the 16th family and the 17th family but have no central cycles, and it is proved that the extension group is a LA- group. The specific methods are as follows: firstly, according to the structure of p- group and central quotient group, some relations which are isomorphic to the 16th and 17th family groups are obtained. Secondly, the existence of the group is judged, the nonexistent group is excluded by the counter-proof method, the existence group is proved by the Schreier group extension theory and the Van Dyek free group theory. Finally, the lower bound of the automorphism group of the extended new group is discussed, that is, the LA- conjecture is verified. To prove that Aut (G) G Aut (G), select a subgroup of Aut (G) R (G) = Ac (G) Inn (G), to transform into proof G R (G), and then get G Aut (G). Finally, it is obtained that some finite aperiodic p- groups with a central quotient order of PQ 6 are LA- groups. That is, in the groups of two families 桅 _ (16) to 桅 _ (17), we find out the LA- group G with central apicyclic and the order of the central quotient group p0 6, such that G / Z (G) (?) H, where H 鈭，
本文编号：2369203
[Abstract]:In the research field of group theory, the optimal lower bound of automorphism group of finite p-group has always been a hot issue. There is a famous LA- conjecture about the best lower bound, that is, let G be a finite aperiodic p-group, and G = pn,n2,. Then there must be a G Aut (G). A group satisfying LA- 's conjecture is called a LA- group. Based on the classification theory of Rodney James's group of order 6, this paper further studies the conjecture of LA-. In this paper, we give a series of finite p- groups whose central quotient groups, generalized by groups of order 6, are isomorphic to the 16th family and the 17th family but have no central cycles, and it is proved that the extension group is a LA- group. The specific methods are as follows: firstly, according to the structure of p- group and central quotient group, some relations which are isomorphic to the 16th and 17th family groups are obtained. Secondly, the existence of the group is judged, the nonexistent group is excluded by the counter-proof method, the existence group is proved by the Schreier group extension theory and the Van Dyek free group theory. Finally, the lower bound of the automorphism group of the extended new group is discussed, that is, the LA- conjecture is verified. To prove that Aut (G) G Aut (G), select a subgroup of Aut (G) R (G) = Ac (G) Inn (G), to transform into proof G R (G), and then get G Aut (G). Finally, it is obtained that some finite aperiodic p- groups with a central quotient order of PQ 6 are LA- groups. That is, in the groups of two families 桅 _ (16) to 桅 _ (17), we find out the LA- group G with central apicyclic and the order of the central quotient group p0 6, such that G / Z (G) (?) H, where H 鈭，
本文编号：2369203
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