| The automorphism groups of the finite p- groups are studied in this thesis. Let group G be a finite p- group. Group G is called LA-group, if the order of the automorphism group of group G devides the order of group G.A finited group is proposed by some given defining relations sets. Then its existent property is researched, further more, Aut(G) and the order of Aut(G) are solved by defining relations sets. The main results of this thesis about the problems of LA-groups are as follows.Result 1 (Proposition 2.1) Let G =pm= bpn= 1, [a, b] = aps, where m, n, s are nonnegtive integers), then s>0, G is a group and G (?)(?), where F = (x,y) is afreegroup, S ={xpm,ypn,[x,y]x-ps}, (?).Result 2 (Proposition 2.2) Let G = pm= bpn = 1, [a, b] = aps, where m, n, s are nonnegtive integers).(1) If m = 2,n≥5,5 = 1,then|Aut(G)|=(p-1)pn+3=(p-1)p|G|;(2) If m-s≤s,then G'≤Z(G),and(i)if n>m,then|Aut(G)|= (p-1)p2m+n+s-1= (p-1)pm+s-1|G|; (ii)if m>n and n + s≥m,then |Aut(G)|=(p-1)p2s+2n-1 .Result 3 (Theorem 2.1) Let G = pm= bpn= cpk = 1, [a,c] = aps, [a,b] = 1, [b,c] = 1, where m,n,k are positive integers and s is a nonnegative integer).(1) If m-s≤s,then c(G) = 2, exp(G')=exp(G/Z(G));(2) If m = k = 2,n≥5,s = 1,thenZ(G)=p>××p>;(3) If k>n>m>s, k + s>2m and m-s2p3m+n+2k+s-2 =(p-1)2p2m+k+s-2|G|.Result 4 (Theorem 2.2) Let G = pm =bpn=cpk = 1, [a,c] = [b,c] = 1, [a, b] = cps, where p is an odd prime and n>k>m>s,n+s>k + m), then|Aut(G)|=(p-1)2p4m+n+k+s-2 = (p-1)2p3m+s-2|G|, i.e. finite group G is LA-group.Result 5 (Theorem 2.3) Let G = pm = bpn = cpk= 1, [a, b] = [a, c] = 1, [b,c] = aps, where p is an odd prime and k>n>m>s, k + s>n + m), then|Aut(G)|=(p-1)pm+3n+k+2s-1(p-1)p2s+2n-1|G|, i.e. finite group G is LA-group.Result 6 (Theorem 2.4) Let G =pm=bpn =cpk =1, [a, b] = [b, c] = 1, [a,c] = bps,where p is an odd prime and m>k>n>s,m + s>k + n ,k +s>2n>, then|Aut(G)|=(p-1)2pm+3n+3k+s-2= (p-1)2ps+n+2k-2|G|, i.e. finite group G is LA-group.
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