Minimal Abelian groups that are not automorphism groups

Abelian groups which can occur as the automorphism groups of a finite group have been investigated by many experts. In 1975, Jonah and Konvisser constructed a group of order p to 8 whose automorphism group is Abelian of order p to 16. In 1983, MacHale showed that the Abelian p-groups which can occur as the automorphism groups of a finite group must be of order greater than p to 1. In this paper, we fix our attention on determining all Abelian p-groups of minimal order which can occur as the automorphism group of a finite group and all groups whose automorphism group is the Abelian p-group of minimal order. We obtain many results about Abelian group which can not occur as the automorphism group of a finite group. Especially, we show that for the group possessing an Abelian automorphism group if it possesses a cyclic commutator subgroup then it must be cyclic itself. By these results we conclude the following: (1) if a non-cyclic group of order p to 7 possesses an Abelian automorphism group, then its automorphism group must be of order p to 12; (2) there is no group whose automorphism group is an Abelian group of order less than or equal to p to 11, where p is not equal to 2. Using the two results and some of Morigi's results we solve some problems given by MacHale in 1983. Our techniques of proof might apply to all Abelian p-groups.