Several Studies Of Non-Cyclic Central Quotients To Isomorphic Groups

In the research field of the theory of group,the lowest boundary of automorphism order of finite p-groups is a hot topic.About the lowest boundary,there is a LA-conjecture:let G be a finite non-cyclic p-group,and|G|= pn,n>2,then |G| divides |Aut?G?|.If the finite non-cyclic group satisfies this conjecture,it is called a LA-group.Based on the Rodney James' classificati-on of order p⁶,this paper will research this conjecture continuously.And it is given that a series of finite p-groups which centers are non-cyclic and central quotients are isomorphic to ?₁₆-?₁₇ whose order are p⁶ and shown that the new obtained LA-groups are verified.At first,based on the structure of p-group and central quotient of p-group,some relations which are isomorphic to the groups of ?₁₆-?₁₇ are gotten.Secondly,We need prove the existence and exclude some nonexistent groups by reduction to absurdity.If the group is existent,its existence will be proved through the theory of extension theory of groups and the method of free group.At last,the lowest boundary of order of automorphism of the new group is considered,namely,testing if or not the new group is a LA-group.In order to prove |G|||R?G?|,a subgroup of Aut?G?will be chose:it is R?G?,i.e.,R?G?= Ac?G?Inn?G?,thus it will be transformed to prove |G|||R?G?|.when |G|||R?G?| is correct,|G|||Aut?G?I is gotten,and it is proved that the new groups of non-cyclic center and central quotients of order p⁶ are all LA-groups.Namely,there exist a class of LA-groups G of non-cyclic center and central quotients of order p⁶ such that G/Z?G????H,H??₁₆-?₁₇.The main results are stated as follow:?1?When H is one of ?₁₆?16?,?₁₆?2211?b,?₁₆?2211?fr,there exist a class of p-groups G of non-cyclic centers and central quotients of order p⁶ such that G/Z?G????H,especially,G are proved to be new LA-groups.?2?When H is one of ?₁₇?16?,?₁₇?2211?f,?₁₇?2211?mr,s,and p = 3,?₁₇?214?c,?₁₇?214?br,?₁₇?214?d,there exist a class of p-groups G of non-cyclic centers and central quotients of order p⁶ such that G/Z?G?= H,especially,G are proved to be new LA-groups.