Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
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1$\begingroup$ You should probably check the work of J. Wiegold $\endgroup$– Geoff RobinsonMar 1, 2015 at 17:01
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$\begingroup$ @GeoffRobinson which one? do you have a reference? $\endgroup$– PabloMar 1, 2015 at 17:02
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1$\begingroup$ He wrote a few papers on "Growth sequences of Groups" $\endgroup$– Geoff RobinsonMar 1, 2015 at 18:42
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2$\begingroup$ it's logarithmic, check Thevenaz's elementary argument: arxiv.org/abs/math/9703201 $\endgroup$– YCorMar 1, 2015 at 19:04
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3$\begingroup$ Essentially a duplicate of this question:mathoverflow.net/questions/187736/… $\endgroup$– Ian AgolMar 1, 2015 at 21:59
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1 Answer
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One has $$1 \leq s_n - \frac{\log(n)}{\log|S|} \leq 2r$$ based on an elementary argument in Remark 1.1 in [Moshe Jarden and Alexander Lubotzky, Random normal subgroups of free pro-finite groups, J. Group Theory 2 (1999) 213-224], where $r$ denotes the minimal number of generators of $S$. By the classification of finite simple groups, we know that $r = 2$.
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$\begingroup$ One may find this article here: tau.ac.il/~jarden/Articles/paper68.pdf $\endgroup$– Ian AgolMar 1, 2015 at 23:30