Some LA-Groups With Non-Cyclic Centres

LA-conjecture is an intuitional conjecture about the lower bound of order of the automorphism group of a finite p-group which is non-cyclic and of order greater than p2, as say|G|||AutG|. For some finite p-groups with special character, the conjecture is true. However, is it true for all finite p-group which is non-cyclic and of order greater than p2, so far there is no definite proof. This article based on the predecessor’s work, constructs some new finite p-groups which accord with| G||| Aut G|. The detailed idea as followed:the thesis is based on the group H of Φ9 to Φ12 with order p6. By using the extension theory, some finite p-groups which satisfy G/Z（G）（?）H are constructed. The existence of structure of the finite p-groups is proved by using the free group theory. Then according to the action of automorphism of the finite p-groups and the relationship|G|||Ac（G）|||R（G）|||Aut（G）|, the finite p-groups satisfy |G||| Aut G|. The content of the chapter is followed:The first chapter expound the background of this thesis in algebra field, the research situation of LA-group at home and abroad, the main wok and results of this thesis.The second chapter summarized the conceptions and the lemma involved in the paper.The third chapter is based on the group H of Φ₉,Φ₁₀,φ₁₁,Φ₁₂, using the extension theory and free group theory to construct some finite p-groups which satisfy G/Z（G）（?）H, and Z（G） is non-cyclic. Furtherly, the finite p-groups G satisfy|G|||Aut G|.