The Generalization Of Davitt’s Conclusion About La-Conjecture

This thesis mainly studies the finite non-cyclic p-groups that are groups of families from Φ1, to Φ10of order p6based on P. Hall iscolinsim concept. For the best lower bound of the order of the automorphism groups of finite p-groups, there is a well-known conjecture, namely LA-conjecture:if G is a finite non-cyclic p-group of order more thanp2, then|G|divides|Aut(G)|, also called G a LA-group. In1980, Davitt proved an important conclusion about LA-group:the finite non-cyclic p-groups G whose central quotients are of order less than p5must be LA-groups. The conclusion was given in recent years: the finite non-cyclic p-groups G whose central quotients are of order p5must be LA-groups. Give all the finite p-groups of families from Φ, to Φ10satisfying central quotients of orders p6in the case of assuming the groups have cyclic centre in reference [42] in2012. On the basis, combined with the characteristics of central and the automorphism group, the paper proves whether their centre is cycle or they are LA-groups. This gives help for proving whether groups who have cycle centre and central quotients of orders p6satisfy LA-groups.The contents of each chapter are as follows:In chapter one, firstly introduce the research background and the present study status at home and abroad about LA-conjecture. Secondly, introduce the main achievement of my published paper during the school.In chapter two, the basic knowledge and the methods involved in the paper are introduced.In chapter three, analyzing the structures of the groups, and combined with the characteristics of them, the paper prove whether the groups have cyclic centre or satisfy LA-groups. If they have cyclic centre, what is the requirement.