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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1You should probably check the work of J. Wiegold – Geoff Robinson Mar 01 '15 at 17:01
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@GeoffRobinson which one? do you have a reference? – Pablo Mar 01 '15 at 17:02
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1He wrote a few papers on "Growth sequences of Groups" – Geoff Robinson Mar 01 '15 at 18:42
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2it's logarithmic, check Thevenaz's elementary argument: http://arxiv.org/abs/math/9703201 – YCor Mar 01 '15 at 19:04
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I would guess that the average number of independent uniformly random elements you need to pick to generate the group also grows logarithmically. – Douglas Zare Mar 01 '15 at 19:54
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possible duplicate of Powers of finite simple groups – Ian Agol Mar 01 '15 at 21:59
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3Essentially a duplicate of this question:http://mathoverflow.net/questions/187736/powers-of-finite-simple-groups – Ian Agol Mar 01 '15 at 21:59
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@IanAgol although you're essentially right I won't vote to close as a duplicate because the question is asked in a much more clear and natural way here than in the linked post – YCor Mar 01 '15 at 22:16
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@Ian Agol: The other question was asking for computations not for asymptotics, in any case this question was not answered. – Andreas Thom Mar 02 '15 at 10:39
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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 $r = 2$.
Andreas Thom
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One may find this article here: http://www.tau.ac.il/~jarden/Articles/paper68.pdf – Ian Agol Mar 01 '15 at 23:30