Some Research Of La-Conjecture

Assume that p is an odd prime. For the lower bound of the automorphism groups of finite p-groups, there is a well-known conjecture, called LA-conjecture, that is, if H is a finite non-cyclic p-group of order greater than p2, then|H|divides|Aut(H)|, it is also said to be a LA-group. Richard M. Davitt, Yu Shuxia, Ban Guining, Li Shiyu and so on, have found out a number of LA-groups by the properties of the center and central quotient of finite p-groups. Based on previous studies, in this thesis we will investigate LA-conjecture for some finite non-cyclic p-groups of orders pn(n>2), mainly focus our attention on non-cyclic and cyclic centers, and obtain several new results. The main contents as follows:(1) It is shown that if G is a finite non-cyclic p-group of order pn where n>2such that the center of G is not cyclic, then the order of the Sylow p-subgroup of Aut(G) is greater than or equal to pmin(8,n) by using counter-evidence method. By the result we conclude that all the groups of orders p8with non-cyclic center are LA-groups. Furthermore, it is proved that for every finite p-group G with non-cyclic center, it is impossible that the order of the automorphism group of G is equal to p7, which extends some known conclusions about the orders of automorphism groups recently.(2) Give some groups G with Z(G) is cyclic and central quotient of order p6and show their existence. In detail, let H be groups of families Φ1to Φ16, and family Φ40of order p6based on P. Hall’s iscolinsim concept and G the groups having cyclic centers of order pn. We first eliminate some groups H of order p6which is not isomorphic to G/Z(G) by using Hall’s identity, formulae of metabelian group for power structure and commutator structure or showing x∈Z(G) to obtain a contradiction. Next, we show there exist groups G by Holder’s theorem, Schreier’s extension, free group and so on, such that G/Z(G)(?)H.