The La-Groups With Non-Cyclic Central Quotients Isomorphic To Some Groups In The Family Of Order P~6

The order of Automorphism of a finite p-group is an important branch of the theory of group. With the calculation of the order of Automorohism group, the problem of the highest and lowest boundary of Automorphism groups was presented by some scholars. And the highest boundary has been solved, yet related to the lowest boundary, there is a conjecture:let G a finite p-group, and|G|≥pn,n> 2, then|G| divides|Aut(G)|. It was named LA-conjecture and a finite non-cyclic group G satisfying this conjecture is called a LA-group. Although the conjecture has not been solved, many beautiful results has been got, based on the results in existence, the dissertation do some research on Φ13-Φ15 whose order are p6, and get some new LA-groups.At first, based on the conclusions proved by Davitt and Ban Guining et, we get:if|G/Z(G)|≤ p5, then G is LA-group. In the paper, let the non-cyclic central quotient of the finite p-group G be isomorphic to H, where H is from Φ13 -Φ15,through different methods, for the special H whose generator is also in Z(G), for group G like this, the order of central quotient of G is less than or equal to p5, then G is a LA-group already. It is equivalent to say there is no finite non-cyclic group G satisfying G/Z(G)≌ H; Secondly, according to the structure of p-group and the non-cyclic center, new groups are deduced, and the existence is proved through the theory of extension of group and free group; At last, the lowest boundary of order of automorphism of the new groups is considered, namely, testing if or not the new groups are LA-groups. It is difficult to verify that|G|divides|Aut(G)|, in the paper, R(G)= Ac(G)Inn(G) will be chose:it is a subgroup of Aut(G), and calculate the order of R(G), and a low boundary of|Aut(G)| is get, when|C|||R(G)| is correct, we get|G|||Aut(G)|, and we proved the new groups are all LA-groups.The main results are stated as follow:(1) When H is one of Φ13(2211)a,Φ13(2211)cr,Φ13(214)a,Φ13(214)b, Φ13(214)d,Φ14(42),Φ14(321),Φ15(2211)a,Φ15(2211)b,Φ15(214),Φ15 (2211)br,s, Φ15(2211)dr, there is no finite p-group G, satisfying G/Z(G)≌ H;(2) When H is Φ13(16),Φ13(2211)b,Φ13(2211)d,Φ13(2211)e,Φ13(2211)f, Φ14(222),Φ15(16), there exist finite p-groups G, satisfying G/Z(G)≌H, and the corresponding structure of G is one of G(13,16),G(13,2211,b),G(13,2211,d), G(13,2211, er), G(13,2211,/), G(14,222), G(15,16), especially, G is proved to be new LA-group.