Here's a wonderful sequence of articles by Tim Silverman depicting modular curves. In particular, he considers actions of the modular group on Z/nZ. https://www.google.com/search?q=site%3Agolem.ph.utexas.edu%2Fcategory%2F+"pictures+of+modular+curves" On Fri, Jul 19, 2013 at 2:40 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Wikipedia mentions this intriguing fact:
PSL(2, p) acts non-trivially on p points if and only if p = 2, 3, 5, 7, or 11.
It says this observation is due to Galois, 1832. More at < https://en.wikipedia.org/wiki/Projective_linear_group#Action_on_p_points>.
Terms are defined at bottom.
QUESTION: Does anyone in math-fun understand this? A heuristic explanation of why non-trivial actions of PSL(2,p) exist for primes 2 <= p <= 11 -- and only for these primes -- would be especially welcome.
That Wikipedia article explicitly describes actions of PSL(2,p) on a set of p points for all primes p <= 11, but does not seem to address the reasons such don't exist for p >= 13. Perhaps the actions that do exist are mathematical coincidences, made more likely by the lowness of the relevant primes?
--Dan _________________________________________________________________________________________ * Just to be clear, for each prime p, PSL(2,p) is the group of formal linear fractional transformations
x -> (ax + b)/(cx + d)
where a,b,c,d are in the field F_p (=the ring Z/pZ), and ad-bc = 1. This is the quotient of SL(2,p) obtained by identifying each matrix with its negative (irrelevant if p = 2, of course).
It's a pleasant calculation to see that #(PSL(2,2)) = 6, and PSL(2,p) = (p^3-p)/2 for p > 2. It's known that PSL(2,p) is simple if and only if p >= 5.
PSL(2,2) == S_3, and PSL(2,3) == A_4. PSL(2,4) == PSL(2,5) == A_5, and PSL(2,7), are the two smallest nonabelian simple groups.
To say that a group G acts on a set X means there is a map
f: G x X -> X , with f(g,x) denoted by gx,
such that for all g and x we have 1x = x and g(hx) = (gh)x. This action is called faithful if gx = x for all x implies that g = 1. The action is called trivial if gx = x for all g and all x.
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