Minimal Abelian automorphism groups of finite groups (Q1915408)

The main result of the paper is the following: Let \(G\) be a finite non-cyclic \(p\)-group, \(p\) odd, for which \(\text{Aut }G\) is Abelian. Then \(p^{12}\) divides \(\text{Aut }G\). This is the best possible lower bound, as \textit{M. Morigi} [Rend. Semin. Mat. Univ. Padova 92, 47-58 (1994; Zbl 0829.20028)] constructed groups \(G\) of order \(p^7\) with \(\text{Aut }G\) Abelian of order \(p^{12}\). For \(p=2\) the minimal order of an Abelian automorphism group of a finite 2-group is 128, for a group of order 64 found by G. A. Miller in 1913.