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FINITE -GROUPS WITH SMALL AUTOMORPHISM GROUP
For each prime $p$ we construct a family $\{G_{i}\}$ of finite $p$ -groups such that $|\text{Aut}(G_{i})|/|G_{i}|$ tends to zero as $i$ tends to infinity. This disproves a well-known conjecture that $|G|$ divides $|\text{Aut}(G)|$ for every nonabelian finite $p$ -group $G$ .