Some Researches On The Orders Of Automorphism Groups

In the research of automorphism group of finite group, what’s worth mentioning is an unresolved famous LA-conjecture:Let G be a non-cyclic finite p-group of order more than p2, then the order of group G divide exactly the order of its automorphism group. The conjecture has been researched for more than half a century, but it has not been completely resolved up to now. In the research of automorphism group, the orders of automorphism groups not only offer theoretical basis for the applying of theoretical physics, quantum mechanics, quantum chemistry, crystallography, cryptography and so on, but also have certain value of completely solving LA-conjecture, so after the classification of simple group was resolved, the calculation of orders of automorphism groups is one basic work in research of automorphism group.In this thesis, firstly, we introduce the backgroud and current researches of the orders of automorphism groups and LA-conjecture. Secondly, we uniformly give all the orders of automorphism groups of groups of order less than or equal to ps and groups of order equal to p6from the twenty-first family to the fortieth family. Moreover, we make a number of corrections for orders of automorphism groups of lower order p-groups offered for mistakes or faults before. Finally, we make some researches of the existence of finite p-group of cyclic center and central quotient of order p6, which lay groundwork for existence of such LA-groups. There are two main points in this thesis:(1) We give the solution formulae of general equations of second degree and two variables in finite field by group theory methods, using the formulae on the basis of relevant number theory knowledge and combining the formulae with matrix equation method, uniformly give all the orders of automorphism groups of groups of order less than or equal to p5and groups of order p6from the twenty-first family to the fortieth family, also make many corrections of orders of automorphism groups of lower order p-groups offered for mistakes before.(2) A number of p-groups are given and their existences are shown, they might be take as either the new LA-groups or the counter examples of LA-conjecture. This completes our study.