Some New La-Groups With P~6 As The Order Of Central Quotients

The finite p-group is one of the historical and most important branches in the theory of finite group, and the study of the finite p-group especially its automorphism group, has always been kept a watchful eye by domestic and overseas scholars. In the dissertation, the research based on the classification of groups of order p6 in Rodney James’paper, by using the method of generators and defining relations, extension theory, free group analyze some new LA-groups. The main work are stated as follows:Chapter One. The background and the current investigation of the finite p-group are introduced, also as the main research of this paper.Chapter Two. The basic knowledge, the definitions and the lemmas involved in the paper are presented, which are prepared for the theorems of chapter three.Chapter Three. Firstly, by using the method of generators and defining relations, some finite non-cyclic p-groups whose central quotients are isomorphic to the group of order p6 in inoclinism families to three and five with the centers are non-cyclic are given; Next, by using the method of extension theory, free group and Van-Dyek’s theorem, the existence of these groups are proved which satisfy the defining relations; Lastly, by combining the Ac(G)≤Aut(G) and the characteristic of their automorphisms, some new LA-groups with central quotients of order p6 and the centers are non-cyclic are obtained.