A result about Abelian automorphism groups (Q1368262)

It is known for odd prime \(p\) that an abelian group of order not more than \(p^7\) can not be the (full) automorphism group of a finite group [see \textit{D. Flannery, D. McHale}, Proc. R. Ir. Acad., Sect. A 81, 209-215 (1981; Zbl 0479.20015)]\ while the order \(p^{12}\) is possible [see \textit{M. Morigi}, Commun. Algebra 23, No. 6, 2045-2065 (1995; Zbl 0836.20019)]. The purpose of this paper is to reduce this gap: It is shown, that no abelian group of order \(p^8\) or \(p^9\) occurs as automorphism group. The proof is by restriction, treating the remaining cases one by one.