Some La-Groups Of Centers Cyclic

If G is a finite non-cyclic p-group of order greater than p2, then|G|divides|Aut(G)|, this is a well-known conjecture, called LA-conjecture, the groups satisfying LA-conjecture are said to be LA-groups. This paper mainly studies whether the non-cyclic p-groups of centers cyclic and central quotients from φ11to φ20based on P.Hall iscolinsim concept are LA-groups or not. In this work we extend the results of LA-conjecture that Davitt, Yu Shuxia and Ban Guining have given, that is, the finite non-cyclic p-group G satisfying p2≤|G/Z(G)|≤p5are LA-groups, to show some p-groups G of central quotient|G/Z(G)|=p6are also LA-groups. All finite non-cyclic p-groups of centers cyclic and central quotient|G/Z(G)|=p6from φ11to φ20have been given, we will proceed on the basis of these works, at the heart of our paper lie the centers cyclic and the subgroups Inn(G)4(G) or46(G)Z(G)(G) of the automorphism groups. Firstly, we judge whether the centers cyclic and what condition are supposed, then in these conditions, we prove the given groups are LA-groups. In a word, in the paper we show the finite non-cyclic p-groups of center cyclic and central quotient from φ11to φ20are LA-groups. The approaches used can be used to the group theory and other subjects. In fact, we only need consider the LA-conjecture in the condition Z(G)≤φ(G) in view of a result due to Hummel, P. Hall has proved that the Aφ(G)Z(G)(G) is a p-group very early. In this paper we prove practically that if|G/Z(G)|=p6, and Z(G) is cyclic, then|Aφ(G)Z(G)(G)|≥|G|. It is clear that|Aφ(G)Z(G)(G)|is a key to study the LA-conjecture.The contents of each chapter are given as follows:In chapter one, we introduce the research background and the research status in china and abroad of LA-groups.In chapter two, the basic knowledge and approaches involved in the paper are introduced.In chapter three, use the commutator and the centers cyclic, find the conditions of the centers cyclic, write them into the congruence equations, then calculate the order of the R=lnn(G)Ac(G) or Aφ(G)Z(G)(G), at last, prove the groups are LA-groups by comparing the order of G with the order of R=Inn(G)4(G) or Aφ(G)Z(G)(G).