| If the finite non-cyclic p-groups G satisfy| G||| Aut(G)| (| G|> p2), then G are called LA-group. Davitt R M, Ban Guining, Yu Shuxia et al have proved that many p-groups are LA-groups. On that basis, this paper is mainly concerned on central quotients from φ31 to φ43 and φ2 based on P. Hall iscolinsim, with the centers cyclic and non-cyclic as two main lines, proving the conclusion that they are LA-groups by studying the best lower bound of automorphism groups. Concretely, there are two main contents:(1) For finite non-cyclic p-groups from φ31 to φ43, some conditions in which the centers cyclic are given by using the knowledge of finite group theory and the fundamental number theory, the order of N-automorphism group Aut(G) of G is calculated by using the congruence equations, parametric and WAG method when Z(G) is cyclic, judging G is a LA-group from checking |G||| Aut(G)|.The results of LA-groups that Ban Guining, Cui Yan and Liu Hailin have given are proceed by these works, that is, the non-cylic p-groups of cyclic centers and central quotients from φ11 to φ30 based on P.Hall iscolinsim are LA-groups. On the basis, the automorphism groups of some finite non-cyclic p-groups whose centers are cyclic and central quotients are isomorphic to the groups of order p6 are proved to be LA-groups.(2) For finite non-cyclic p-groups from φ2, some groups H of order p6 which is not isomorphic to central quotient G/Z(G) are excluded by using formulate of meta-commutative group for power structure and commutator structure first, then some groups G which satisfy G/Z(G)≡H are certified by using Schreier’s extension, the theory of free group and so on, at last, the groups are proved to be LA-groups by combining with the characteristic of their automorphisms. The conclusion lay a foundation for proving that automorphism groups of some finite non-cyclic p-groups whose centers are non-cyclic and central quotients are isomorphic to the groups of order p6 are LA-groups in the future. |
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