Finding conjugate stabilizer subgroups in PSL(2; q) and related groups

Details Finding conjugate stabilizer subgroups in PSL(2; q) and related groups Finding conjugate stabilizer subgroups in PSL(2; q) and related groups

2008-09-15

arxiv.org/abs/0809.2445v5

Physics

Quantum Physics

10 pages, added details of representation theory for non-prime q

Scientific paper

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).

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