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Does someone know whether the order of the automorphism group of a general p-group of order $p^n$ is bounded from above by $(p^n)^2 $? (Every element can possibly be transferred to one of other $p^n$ elements)...

If this fact is incorrect, is it possible to deduce a bound on the order of such an automorphism group if our p-group is finitely generated?

Thanks in advance

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    $\begingroup$ See this previous question: mathoverflow.net/questions/68109/…. The order of the automorphism group of a group of order $p^n$ divides $(p^n-1)(p^n-p)\cdots(p^n-p^{n-1})$, and the bound is sharp, achieved by the elementary abelian $p$-group of rank $n$. $\endgroup$
    – Arturo Magidin
    Commented May 30, 2012 at 16:12
  • $\begingroup$ So, is it true that the order of the aut group if $O(p^{2n})$? $\endgroup$
    – jason mfash
    Commented May 30, 2012 at 16:38
  • $\begingroup$ The bound Arturo gave is not $O(p^{2n})$ but $O(p^{n^2})$. $\endgroup$
    – Andreas Blass
    Commented May 30, 2012 at 17:12

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For a specific counterexample to the fact that $p^{2n}$ is not an upper bound, take the elementary abelian group of order $p^3$. It has automorphism group of order $(p^3-1)(p^3-p)(p^3-p^2)$ (pick a basis; the first basis vector can go to any nonzero vector; the second to any vector not in the linear span of the first; the third to any vector not in the linear span of the first two images). For $p=3$, this gives $11232$ automorphisms, larger than $(3^3)^2 = 729$.

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  • $\begingroup$ So, is it true that the order of the aut group if $O(p^{2n})$? $\endgroup$
    – jason mfash
    Commented May 30, 2012 at 16:39

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