On finite $p$-groups with abelian automorphism group
We construct, for the first time, various types of specific non-special finite $p$-groups having abelian automorphism group. More specifically, we construct groups $G$ with abelian automorphism group such that $γ_2(G) < \mathrm{Z}(G) < Φ(G)$, where $γ_2(G)$, $\mathrm{Z}(G)$ and $Φ(G)$ denote the commutator subgroup, the center and the Frattini subgroup of $G$ respectively. For a finite $p$-group $G$ with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) $\mathrm{Z}(G) = Φ(G)$ is elementary abelian; (ii) $γ_2(G) = Φ(G)$ is elementary abelian, where $p$ is an odd prime. We construct examples to show the existence of groups $G$ with elementary abelian automorphism group for which exactly one of the above two conditions holds true.