Let p be an odd prime, and let G denote PSL(2,F_p). (Cf. < http://en.wikipedia.org/wiki/Projective_special_linear_group >.) (Each such G is simple, of order #G = (p^3-p)/2.) 1. What's the least n for which the orthogonal group O(n) contains G as a subgroup ? Denote this n by Or(p). 2. What's the least n for which the symmetric group S_n contains G as a subgroup ? Denote this n by Sy(p). (It's clear that Or(p)+1 <= Sy(p) <= #G since any group permutes itself faithfully, and any S_n embeds in O(n-1) via isometries of the regular (n-1)-simplex.) --Dan