# What is the growth of the rank of a power of a finite simple group?

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?

• You should probably check the work of J. Wiegold – Geoff Robinson Mar 1 '15 at 17:01
• @GeoffRobinson which one? do you have a reference? – Pablo Mar 1 '15 at 17:02
• He wrote a few papers on "Growth sequences of Groups" – Geoff Robinson Mar 1 '15 at 18:42
• it's logarithmic, check Thevenaz's elementary argument: arxiv.org/abs/math/9703201 – YCor Mar 1 '15 at 19:04
• Essentially a duplicate of this question:mathoverflow.net/questions/187736/… – Ian Agol Mar 1 '15 at 21:59

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 $r = 2$.