Question

There is a theorem that says that if $p$ is a prime and $G$ is a $p$-group with $|G| = p^{n}$, $|Aut(G)|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$.
This theorem is sharp, since $\Pi_{k=0}^{n-1} (p^{n}-p^{k}) = |GL(n,p)| = |Aut(E)|$, where $E$ is an elementary abelian group of order $p^{n}$.
The proof I know is by proving the $p$-part and the $p^{'}$-part of the divisibility separately.
The $p^{'}$-part of the divisibility boils down to the integrality of the binomial coefficient analogues which count decompositions of a vector space over $\mathbb{Z}/(p)$ into two subspaces whose sum is the space and whose intersection is $0$.
The $p$-part of the divisibility is the divisibility statement for the order of a Sylow $p$-subgroup of $Aut(G)$, and it uses induction and the fact that a $p$-group will have fixed points whenever it acts on a set whose cardinality is a nonmultiple of $p$.
Is there a book that covers this theorem? If so, how far (and where) does the book run with it?

Answer 1

This is just Burnside's basis theorem. See for instance theorem 12.2.2 on page 178 of M. Hall, Jr.'s textbook on the Theory of Groups. The original reference for the phrasing in terms of automorphisms is from P. Hall (1933).

As far as where to go from it, this is roughly how automorphism groups of p-groups are calcuated in O'Brien (1992), Eick–Leedham-Green–O'Brien (2002) and the AutPGrp package of GAP. This is often useful in understand fusion systems, where the p-core of automorphism groups is under good control, and so the GL(n, p) part is the primary interest.

Another application (known to the OP, but interesting enough to describe clearly) is a result of Burnside (1905) classifying for which powers b = b(p, a, q) there is a group of order p^aq^b with no non-identity normal p-subgroup: the classification is based simply on the orders of the automorphism groups of the p-subgroups. Burnside had an error in his analysis of the associated arithmetical condition that was corrected in Coates–Dwan–Rose (1976). Burnside's result was generalized in Glauberman (1975) and Bialostocki (1975, 1987). Many of these and further results are based on analyzing nilpotent p′-subgroups of GL(n, p), resting ultimately on the fact that that every p′-subgroup of the automorphism group of a p-group of rank n is isomorphic (including in some sense, its action) to a subgroup of GL(n, p).

• Burnside, W. On groups of order p^αq^β Lond. M. S. Proc. (2) 1, 388-392 (1904). JFM35.0162.01 DOI:10.1112/plms/s2-1.1.388

• Burnside, W. On groups of order p^αq^β (second paper). Lond. M. S. Proc. (ser 2) 2, (1905) 432-437. JFM36.0198.02 DOI:10.1112/plms/s2-2.1.432

• Hall, P. A contribution to the theory of groups of prime-power order. Proc. Lond. Math. Soc., Ser. 2, 36, (1933) 29-95. Zbl0007.29102 DOI:10.1112/plms/s2-36.1.29

• Glauberman, G. On Burnside's other p^aq^b theorem. Pacific J. Math. 56 (1975), no. 2, 469–476. MR412269 URL:euclid.pjm/1102906371

• Bialostocki, Arie. On products of two nilpotent subgroups of a finite group. Israel J. Math. 20 (1975), no. 2, 178–188. MR407148 DOI:10.1007/BF02757885

• Coates, Martin; Dwan, Michael; Rose, John S. A note on Burnside's other p^αq^β theorem. J. London Math. Soc. (2) 14 (1976), no. 1, 160–166. MR419594 DOI:10.1112/jlms/s2-14.1.160

• Bialostocki, Arie. On the other p^αq^β theorem of Burnside. Groups–St. Andrews 1985. Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 1, 41–49. MR879428 DOI:10.1017/S0013091500017946

• O'Brien, E. A. Computing automorphism groups of p-groups. Computational algebra and number theory (Sydney, 1992), 83–90, Math. Appl., 325, Kluwer Acad. Publ., Dordrecht, 1995. MR1344923

• Eick, Bettina; Leedham-Green, C. R.; O'Brien, E. A. Constructing automorphism groups of p-groups. Comm. Algebra 30 (2002), no. 5, 2271–2295. MR1904637 DOI10.1081/AGB-120003468

Answer 2

Berkovich's Groups of Prime Power Order, Volume 1 would be a good reference, especially chapter $6$. Theorem $6.9$ generalizes what you mentioned above by calculating directly $|\mathrm{Aut}(G)|$.