A Class Of LA-Groups With Non-Cyclic Centres And P~6 As The Order Of Central Quotients

A finite p-group G is called a LA-group, if the order of G divides the order of the automorphism group of G. In the thesis, the research of LA-conjecture based on the Rodney James’classification of groups of order p6 is continued. At first, a new class of p-groups which centers are non-cyclic and central quotients are isomorphic to the groups of Φ6 to Φ8 are obtained, by using the defining relations of free group and the extension theory of groups; Secondly, the existence of these groups which satisfy defining relations through extension theory of groups and the method of free group are showed; At last, use the characteristic of their automorphisms and the fundamental number theory method to calculated the order of N-automorphism group Aut N(G) (that is, Ac (G), R(G)) of G, then prove these p-groups are LA-groups.The main results of this paper are given as follows:(1) In the sixth family Φ6, there is a class of LA-groups which centers are non-cyclic and central quotients of order p6 such that G/Z(G)≌H while H= Φ6(2211)br,Φ6(214)a, Φ6(214)br,Φ6(16),(D6(2211)g,Φ6(2211)hr and Φ(214)d;(2) In the seventh family Φ7, there is a class of LA-groups which centers are non-cyclic and central quotients of order p6 such that GIZ(G)≌H while H=Φ7(16),Φ7(2211)br,Φ7(2211)f,Φ7(214)f and Φ7(214)g;(3) In the eighth family Φ8, there is a class of LA-groups which centers are non-cyclic and central quotients of order p6 such that G/Z(G)≌ H while H= Φ8(33) andΦ8(222).