Let $G$ be a finite abelian $p$group. What is known about the minimal number of generators of a $p$sylow of $Aut(G)$? is it bounded in terms of $d(G)$ the minimal number of generators of $G$ (and perhaps $p$)?
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For the special case that $G=(\mathbb{Z}/p\mathbb{Z})^n$, this is true. Then we have $d(G)=n$ and $Aut (G)=GL(n,p)$. In the article of A. Patterson, "The minimal number of generators for $p$subgroups of $GL(n, p)$" of $1974$ it is shown that any $p$subgroup of $\text{GL}(n,p)$, where $p$ is an odd prime, can be generated by ${\textstyle\frac 1{4}}n^2$ elements. So the bound is $\frac{1}{4}d(G)^2$ in this case. 

