A New Kind Of LA-Group And Minimal Abelian Group That Is Automorphism Group

On the estimate of the best lower bound of the orders of automorphism groups of finite p-groups, there is a famous LA-conjecture, namely, let G be a finite non-cyclic p-group of order greater than p2, then|G||| Aut(G)|. The research of the problem could be traced back to at least the year of 1955, so far it has not been solved. This thesis constructs a new kind of LA-group based on the groupΦ7(2111)a=(α,α1,α2,α3,β][αi,α]=αi+1, [α1,β]=α3αp,α1p=αi+1P=βp=1(p> 3,i=1,2)) of order p5.In the study of automorphism groups of finite groups, there is a Miller problem, namely, solving the automorphism group equation B = Aut(X). It is very difficult to solve the equation. Professor Ban showed the abelian p-group of the smallest order which can occur as the automorphism group of a finite group is of order p12 when p is an odd prime [54]. This thesis further shows this abelian p-group of order p12 is an elementary abelian group.This thesis consists of the following three chapters:Chapter one analysis the research history and the present situation of LA-conjecture and solving the automorphism group equation B = Aut(X), then brieflv introduces the main research of this thesis.Chapter two generalizes the above group of order p5 by applying ex-pansion theory and free group theory, and obtains a new kind of group G=(a,a1,a2,a3,b|[a1,a]=a2,[a2,a]=a3,[a1,b]=apn=a3spt,a1pm a2pm2=a3pm3=bpm=1,m3≤m2≤min{m1,n),,m3≤m+t,(s,p) 1,0≤t<m3>,then studies some properties of group G,and shows G is an LA-group by a new method.Chapter three proves Aut(G)must be an elementary abelian group if it is an abelian p-group of order p12 with exclusive method. Here, p is an odd prime.