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

TL;DR

This paper reduces certain hidden subgroup problems in groups like PSL(2; q) to solvable problems in affine groups, providing the first positive results for HSP in some finite simple groups, using group-theoretic methods.

Contribution

It introduces a reduction of HSP in PSL(2; q) and related groups to affine group problems, highlighting a novel approach in finite simple group analysis.

Findings

Reduced HSP in PSL(2; q) to affine group problems

First positive results on HSP in finite simple groups

Utilized group actions on projective space

Abstract

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).