Graph isomorphism is in SPP

We show that graph isomorphism is in the complexity class SPP and hence it is in ⊕P (in fact, it is in Mod_kP for each k≥2). We derive this result as a corollary of a more general result: we show that a generic problem FIND-GROUP has an FP SPP algorithm. This general result has other consequences: for example, it follows that the hidden subgroup problem for permutation groups, studied in the context of quantum algorithms, has an FP^SPP algorithm. Also, some other algorithmic problems over permutation groups known to be at least as hard as graph isomorphism (e.g. coset intersection) are in SPP, and thus in Mod_kP for each k>2.