In the following post by DavidLHarden : See Here
He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n $ , then $|AutG| $ divides $ \Pi_{k=0}^{n-1} (p^{n}-p^{k}) $ " .
I can't find any reference for this theorem ,
Does someone know of any reference for this fact?
Thanks in advance
I don't have a reference, but here's the next best thing: a proof.
First, let's fix some notation. Let $P$ be a p-group, $G$ it's group of automorphisms, and $\Phi(P) = P^p[P,P]$ it's Frattini subgroup. Define inductively $\Phi^k(P)$ as $\Phi(\Phi^{k-1}(P)).$
The subgroups $\Phi^k(P)$ form a decreasing chain of characteristic subgroups which exhaust $P.$
Let $$G_k := \ker(G \rightarrow Aut(P/\Phi^k(P)))$$ and $$G_k' := \ker(G \rightarrow Aut(\Phi^k(P)/\Phi^{k+1}(P))).$$
Then the subgroups $G_k$ form an decreasing chain of normal subgroups of $G$ which exhaust $G.$
Let $d_k = \dim_{\mathbb{F}_p}(\Phi^k(P)/\Phi^{k+1}(P)).$ The group $P$ can be generated by $d_0$ elements. Choose a generating set $g_1 ... g_{d_0}$ and consider the map from $G_k \cap G_k'/G_{k+1}$ to $(\Phi^k(P)/\Phi^{k+1}(P))^{d_0}$
given by
$$\sigma \mapsto (\sigma(g_i)g_i^{-1})_{i=1}^{d_0}.$$
This map is an injective group homomorphism.
On the other hand $G_k/(G_{k} \cap G_{k}')$ injects into $Aut(\Phi^k(P)/\Phi^{k+1}(P)) \cong GL_{d_i}( \mathbb{F}_p).$
Note that if $p^n = |P|$ and $r = max\{d_i : d_i \neq 0\},$ then $\displaystyle\sum_{i=0}^r d_i = n.$ It follows that the order $$|G| = \displaystyle\prod_{k=0}^{r}|G_k/G_{k+1}| = \displaystyle\prod_{k=0}^{r} |(G_k \cap G_k')/G_{k+1}||(G_k \cap G_k')/G_{k+1}|$$ divides
$$\displaystyle\prod_{s=0}^{d_0-1} (p^{d_0} - p^s)\displaystyle\prod_{k=1}^{r} p^{d_kd_0}\displaystyle\prod_{s=0}^{d_k-1} (p^{d_k} - p^s)$$
which divides
$$\displaystyle\prod_{k=0}^{n-1} (p^n - p^k).$$
As @DavidLHarden explains in the link that you gave, this theorem is proved by attending to the $p$-part and $p'$-part separately.
For the $p'$-part the result follows from the following theorem of Burnside:
Let $\psi$ be a $p'$-automorphism of the $p$-group $P$ which induces the identity on $P/\Phi(P)$. Then $\psi$ is the identity automorphism of $P$.
This is the result that Geoff refers to in his comment above. It is discussed and proved in Section 5 of Gorenstein's Finite Groups, specifically Theorem 1.4 of that section.
I do not know of a reference for the $p$-part of the proof. You should certainly look at the paper by Neumann that Geoff mentions, however if I understand that proof correctly it only proves your bound for $|Out P|$, rather than $|Aut P|$. On the other hand Neumann is considering a much more general setting than just $p$-groups.
In fact, if $|G|=p^n$ and $d(G)=d$, then $|\mathrm{Aut}(G)|$ divides the number $$(p^n-p^{n-d})(p^n-p^{n-d+1}) ... (p^n-p^{n-1}).$$ This result is due to P. Hall (1933)
Proof. The above product is the number of minimal bases of $G$. However, that number is a multiple of $|\text{Aut}(G)|$. Indeed, if $\cal B$ is the set of all bases of $G$, then all $\text{Aut}(G)$-orbits on $\cal B$ have the same cardinality $|\text{Aut}(G)|$.
It's actually easy to put a bound on $Aut(P)$ for some finite $p$-group $P$, simply using Burnside's basis theorem, which says every basis for the elementary abelian group $P/\Phi(P)$ corresponds to a minimal generating set for the group $P$. Clearly, any element of $Aut(P)$ must take one minimal generating set to another.
So how many minimal generating sets are there? Well, if $|\Phi(P)| = p^d$, and $|P/\Phi(P)|=p^e$, then a minimal generating set consists of $e$ elements. There are $$ \prod_{k=0}^{e-1} (p^e-p^k) $$ different bases of $P/\Phi(P)$ (as a vector space of dimension $e$). Each such element really represents a coset of $\Phi(P)$, which contains $p^d$ elements; that is, for such a given basis, each basis vector has $p^d$ choices up in $P$; all told, then, there are $$ p^{de} \prod_{k=0}^{e-1} (p^e-p^k) $$
minimal generating sets. Of course, $Aut(P)$ acts on these freely (as it is defined by what it does to a generating set), so its order divides that number.
In post 6 is presented a weak result (it attained only for elementary abelian $P$). The Hall's result in post 3 is attained by many groups. It may be improved only in the case if we have an additional information on $P$.
