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Suppose I have a group $G$ and a subgroup $H< G,$ where $|H| = k, |G| = k l.$ Let $A_H < Aut(G) = \{\phi \left| \phi(H) = H\right.\}.$ The question is: is there any decent (where you can define "decent" in your favorite way) universal upper bound on the index of $A_H$ in $Aut(G)$ in terms of $k, l?$

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Can I define decent such that $\left(\begin{array}{c} kl \\ k\end{array}\right)$ is decent? –  Will Sawin Nov 8 '12 at 2:12
    
Well, this is decent in some circles, but blows up rather badly with both $k$ and $l.$ Are you thinking of some example where this is close to being achieved? –  Igor Rivin Nov 8 '12 at 3:54
    
I would certainly expect that there was not one! My comment is intentionally very silly. In fact, I can't think of an example where $A_H\geq |G|$. Can you? –  Will Sawin Nov 8 '12 at 5:07
    
Oh I guess if you take $k=2^n$, $l=2^n$, $G = \mathbb Z/2^{2n}$, $H=\matbb Z/2^{n}$ you get a pretty big number, about $2^{n^2}$. –  Will Sawin Nov 8 '12 at 5:09
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The number of generators of $H$ is at most $\ln_2 k$. This gives an upper bound of $(kl)^{\ln_2 k}$, or $e^{O( (\ln k)^2+ \ln k \ln l)}$, which is closer to the lower bound of $e^{\Theta(\ln k \ln l)}$ you get from vector spaces over $\mathbb F_2$. –  Will Sawin Nov 8 '12 at 5:58

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