A Class Of La-groups With Non-cyclic Central Quotients Isomorphic To The Groups Of Order P~6 Of Nineteenth Family

Group theory is a milestone in the history of mathematics.Since 1829 Galois found groups and gave the sufficient and necessary conditions of an equation to be solved by radical the group theory now has been the development of turn the world upside down,and is considered to be the basic tools of mathematics and many other applications.A finite non cyclic group is called LA-group,if |G|||Aut?G?| and |G| = pn,n>2.In the thesis,the research of LA-conjecture based on the Rodney James' classification of groups of order p6 is continued,which accord to the concept of isoclinism.At first,a new class of p-groups which centers are non-cyclic and central quotients are isomorphic to the groups of p6-group of nineteenth family are obtained,by using the commutator structures and power structures of group;Secondly,the existence of these groups which satisfy defining relations through extension theory of groups and the method of free groups are showed;At last,the characteristic of their automorphisms and the fundamental number theory method are used to calculated the order of N-automorphism group AutN?G?of G,then it is proved that these p-groups are LA-groups.The main results of this paper are given as follows:In the nineteenth family,there is a class of LA-groups G which centers are non-cyclic groups and central quotients are groups of order p6 such that G/Z?G????H where H = ?₁₉?2211?br,?₁₉?2211?cr,s,?₁₉?2211?d0,0,0,?₁₉?2211?dr,s,t,?₁₉?2211?er,?₁₉?2211?fr,s,?₁₉?2211?gr,0,0,?₁₉?2211?gr,s,t,?₁₉?214?a,?₁₉?214?er and?6?16?.