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.