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answered Generating a group by randomly sampling generators
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comment Generating a group by randomly sampling generators
A similar linear upper bound is $2^n k$, which is tighter for small $k$. These are all fantastic, but do you think it might be possible to get a strictly concave lower bound? The linear ones are not quite strong enough for my purposes, unfortunately.
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comment Generating a group by randomly sampling generators
Yes, of course. (I read "with" instead of "without".) In fact, $k \le 3^n$, the number of full elements in $G^n$. Then $N_{3^n} = 4^n \gg 3^n$. In that case, it seems the bound is rather loose, no? Plus one regardless!
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comment Generating a group by randomly sampling generators
I lost you after the first line... $N_k \le |G|^n = 4^n$ for all k, so the bound your wrote can't always hold.