【摘要】：在群論的研究領域中,有限p-群的自同構群階的最佳下界一直是一個熱點問題,關于最佳下界有一個著名的LA-猜想,即設G是有限非循環(huán)p-群,|G| =pn,n2,則一定有|G|||Aut(G)|.滿足LA-猜想的群稱為LA-群.本文立足于Rodney James的p~6階群的分類理論基礎上,進一步展開對LA-猜想的研究工作.本文擬給出了一系列由p~6階群推廣的中心商群同構于第16家族和第17家族但中心非循環(huán)的有限p-群,由此驗證擴張群是否為LA-群.具體方法如下:首先,根據(jù)p-群和中心商群的結構,得出一些同構于第16家族和第17家族群滿足的關系式;其次,判斷該群的存在性,通過反證法排除不存在的群,存在的群則利用Schreier群擴張理論和Van Dyek自由群理論證明其存在性;最后,討論擴張后的新群的自同構群的下界,即驗證 LA-猜想.為證|G|||Aut(G)|,選取Aut(G)的一個子群R(G)= Ac(G)Inn(G),從而轉化成論證|G|||R(G)|,進而得到|G|||Aut(G)|,最終得到若干中心非循環(huán)且中心商群的階為p~6的有限非循環(huán)p-群是LA-群.即在Φ_(16)到Φ_(17)這兩個家族的群中,找出存在中心非循環(huán)且中心商群的階為p~6的LA-群G,使得G/Z(G)(?)H,其中H∈Φ_(16)-Φ_(17).本文主要結果:(1)當H=Φ_(16)(16),Φ_(16)(2211)b,Φ_(16)(2211)fr時,存在中心非循環(huán)的有限p-群G,使得中心商群G/Z(G)(?)H,并且G是LA-群;(2)當H=Φ_(17)(16),Φ_(17)(2211)f,Φ_(17)(2211)mr,mr,s時,及p =3,H=Φ_(17)(214)C,Φ_(17)(214)br,Φ_(17)(214)d時,存在中心非循環(huán)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，

本文編號：2369203