Abelian permutation group problems and logspace counting classes

The goal of this paper is to classify abelian permutation group problems using logspace counting classes. Building on McKenzie and Cook's [MC87] classification of permutation group problems into four NC1 Turing-equivalent sets, we show that all these problems are essentially captured by the generalized logspace modclass ModL, where ModL is the logspace analogue of ModP (defined by Kobler and Toda [KT96]). More precisely, our results are as follows: 1. For abelian permutation groups, the problems of membership testing, isomorphism testing and computing the order of a group are all in ZPL(ModL), and are all hard for ModL under logspace Turing reductions. 2. The problems of computing the intersection of abelian permutation groups, and computing a generator-relator presentation for a given abelian permutation group are in FLModL/poly. Furthermore, the search version of membership testing is also in FLModL/poly.