Finite p-groups with small automorphism group

Abstract: For each prime $p$ we construct a family ${G_i}$ of finite $p$-groups such that $|\Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|\Aut(G)|$ for every non-abelian finite $p$-group $G$.