3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ You should probably check the work of J. Wiegold $\endgroup$ – Geoff Robinson Mar 1 '15 at 17:01
  • $\begingroup$ @GeoffRobinson which one? do you have a reference? $\endgroup$ – Pablo Mar 1 '15 at 17:02
  • 1
    $\begingroup$ He wrote a few papers on "Growth sequences of Groups" $\endgroup$ – Geoff Robinson Mar 1 '15 at 18:42
  • 2
    $\begingroup$ it's logarithmic, check Thevenaz's elementary argument: arxiv.org/abs/math/9703201 $\endgroup$ – YCor Mar 1 '15 at 19:04
  • 3
    $\begingroup$ Essentially a duplicate of this question:mathoverflow.net/questions/187736/… $\endgroup$ – Ian Agol Mar 1 '15 at 21:59
8
$\begingroup$

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 $r = 2$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.