Finding conjugate stabilizer subgroups of PSL(2;q)

We reduce a case of the hidden subgroup problem (HSP) in several families of finite groups of Lie type, such as SL(2; q), PSL(2; q), and PGL(2; q), to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an “almost” 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup. Our observation is purely grouptheoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL(2; q).