  We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q),
and PGL(2; q), three related families of finite groups of Lie type, 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, which
is also the stabilizer subgroup of an element of projective space. Our
observation is mainly group-theoretic, 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).
