The Generalization Of One Of Davitt’Important Conclusion And Releavant Research

This thesis mainly studies the p-groups whose central quotients have orders p5and also discusses some finite non-cyclic p-groups having cyclic centre and central quotients of orders p6. Professor Ban and other authors show that groups having cyclic centre and satisfying central quotients of orders p5are also LA-groups, partly promote Davitt’the important conclusion about LA-conjecture in1980, that is, the non-cyclic p-groups G whose central quotients are of order less than p5must be LA-groups. So the main contents and results of this thesis constructed as follows:(1) In the thesis we first determine all p-groups satisfying that the centers are non-cyclic and central quotients of orders p5, and then show their existence by extension theory of group and the method of free group. Next we prove these p-groups are LA-groups by the characteristic of their automorphisms, and also completely generalize the Davitt’s important result for LA-conjecture in1980.(2) In order to generalize the important conclusions of this thesis in the future, we study p-groups which centers are cyclic and central quotients of orders p6, then a number of new p-groups of cyclic centre and central quotients of orders p6are obtained by extension method based on the groups in inoclinism families seventeen to twenty three and forty one in Rodney James’ paper in1980, which are either LA-groups or some counter examples of LA-conjecture.