Some La-Groups With Cyclic Centres And P~6as The Order Of Central Quotients

In the theory of finite group, the p-group is one of the most fundamental and important branches, and the research of the finite p-group, especially its automorphism group, has always been kept a watchful eye by domestic and overseas scholars. About the order of the automorphism group of finite noncyclic p-group, there is a well-known conjecture, that is to be said LA-conjecture:If G is a finite noncyclic p-group,｜G｜=p",n>2, then｜G‖｜Aut(G)｜,G is called LA-group if it satisfies LA-conjecture. LA-conjecture has been studied for more than half a century and its some important results have been obtained, but it has been still opened. In the dissertation, the research of LA-conjecture based on the classification of groups of order p6and some finite noncyclic p-groups is continued, the main work is focused on automorphism groups of some finite noncyclic p-groups whose centers are cyclic and central quotients are isomorphic to the groups of order p6, the main contents and achievements are stated as follows:The main contents:In the paper, the centers Z(G) and the automorphism groups Aut(G) of some finite noncyclic p-groups is studied G. By the knowledge of finite group theory and the fundamental number theory, the condition that Z(G) is a cyclic group is got. While Z(G) is cyclic, use the parametric method and the solution of congruence equations to calculated the order of N-automorphism group AutN (G)(that is, Ac(G), R(G),AΦ,(G)Z(G)(G)) of G, then check whether G satisfies the conditions of LA-conjecture, that is, whether G is a LA-group. In the following is the detailed process:In order to show that G is a LA-group, need only consider the PN-group G in view of a result due to Exarchakos T. Since AC(G)≤R(G)≤Aut(G) and Ac(G)≤AΦ(G)Z(G)(G)≤Aut(G),｜Ac(G)｜‖(G)｜‖Aut(G)｜and｜Ac(G)｜‖AΦ(G)Z(G)(G)｜‖Aut(G)|.Again Ac(G), R(G),AΦ(G)Z(G)(G) are all p-groups, so｜G｜≤｜Ac(G)｜, or｜G｜≤｜R(G)｜, or｜G｜≤｜AΦ(G)ZG)(G)｜,thus｜G‖｜Aut(G)｜,that is, G is a LA-group.The main achievements:It is gained from the research of automorphism group of some finite noncyclic p-groups that some finite noncyclic p-groups whose centers are cyclic and central quotients are of order p6are LA-groups, that is, if a finite noncyclic p-group G satisfies that center Z(G) is cyclic,｜Z(G)｜=pn,n>1, and central quotient G/Z(G) is isomorphic to the groups H of orders p6, where H is the group of Φ21to Φ30,then G is a LA-group.