Upper bound of |Aut(G)| for a p-group If G is a p-group which is finitely generated with order p^n then what is the upper bound of |Aut(G)|.
 A: There is a theorem by P. Hall which states the following. Let $G$ have a minimal generating set consisting of $d$ elements. Then the order of $\mathrm{Aut}(G)$ divides $p^{d(n-d)}|\mathrm{GL}(d,p)|$. This follows from Burnside's basis theorem for $p$-groups.
A: An upper bound is $|{\rm GL}(n,p)|$, and if $G$ is not elementary Abelian, the bound is better. This is well-known, but I am not sure of a reference for the first proof in the literature: here is the argument: Let $\{x_{1},x_{2},\ldots, x_{t}\}$ be a minimal generating set for $G$, and let $\phi: G \to G$ be an automorphism. Then $\phi$ is clearly specified by $\{\phi(x_{1}),\ldots, \phi(x_{t}) \}$, and the latter set is also a minimal generating set.
Let $X_{1} = \{1_{G} \}$ and $X_{i} = \langle \phi(x_{1}),\ldots,\phi(x_{i-1}) \rangle$ for $i > 1.$ Then $\phi(x_{i}) \in G \backslash X_{i}$ for $1 \leq i \leq t$, and $|X_{i}| \geq p^{i-1}$ for each $i$. Hence $|{\rm Aut}(G)| \leq \prod_{i=1}^{t} (p^{n}-p^{i-1})$, while we certainly have $t \leq n.$
