Finite Nilpotent Groups With Automorphism Groups Of Order 16p~n (n=2,3,4)

This paper discusses the structures of nilpotent group G with fixed orders of automorphism groups,In the process.It is well known that nilpotent group can be decomposed into a direct product of all its Sylow pi-subgroups, where|G|= p1α1p2α2…ptαt,further|Aut(G)|=Πi=1t|Aut(Pi)|(Pi∈Sylpi(G).i=1,…,t).The author discusses the number of prime factors of the order of group G. then the author separately discusses the orders of automorphism groups of Pi to determine the structure of the group G. At last.the author come to the classification of all the finite nilpotent groups when the order of automorphism groups having orders 16pn( n=2,3,4) where p being odd primes.The first part is about finite groups with its automorphism group having or-der 16p3(p is a prime) nilpotent group, all such nilpotent group are in 80 distinct isomorphism classes.Consequently, a classification of nilpotent groups with its automorphism group order for 16 p2(p is a prime) is given, totally 41 distinct isomorphism classes.The second part is about classification of nilpotent groups with its Automor-phism group order 16p4(p is a prime). the author found out all 103 isomorphism classes of such kind groups.