Somehow [this question](https://mathoverflow.net/questions/15444/the-phenomena-of-eventual-counterexamples) made me think of instances of small exceptions in general, and I remembered the statement I heard once that $S\_5,A\_6,S\_6,A\_7,A\_8,S\_8$ are the only instances of symmetric/alternating groups that are not quotients of $Z/2Z\*Z/3Z=PSL\_2(Z)$ (see [this MathSciNet entry](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=TI&pg7=ALLF&pg8=ET&review_format=html&s4=Wiegold&s5=generators&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=7&mx-pid=280580), which I just found). Does anyone have an idea of a conceptual explanation for this fact? 


**Edit:** I also find [this article](http://www.ams.org/bull/1898-04-04/S0002-9904-1898-00473-1/supplementary-information/home.html) which mentions the same result. It's quite interesting that the positive part (for all $n>8$ these groups are quotients) is proved using Bertrand's postulate. I think it's cool that Bertrand's postulate can be used for group theory.



Symmetric groups which are not quotients of Z/2Z*Z/3Z
Yes, this is well known. In fact, there is a name for such groups - this is a (2,3)-generation property. And yes, by now there is a conceptual understanding why all sufficiently large finite simple groups have this property - the basic ideas are outlined in [this](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1405944&loc=fromrevtext) helpful MathSciNet review gently explaining the major breakthrough by Liebeck and Shalev (1996). There are more recent developments in the field, both in the asymptotic direction and in the explicit construction, such as figuring out which $PSL(n,q)$ are (2,3)-generated - see papers by Tamburini, etc. - the literature is too big to be reviewed here. 


