Abstract
The Group Isomorphism problem consists in deciding whether two input groups G_1 and G_2 given by their multiplication tables are isomorphic. We first give a 2-round Arthur-Merlin protocol for the Group Non-Isomorphism problem such that on input groups (G_1,G_2) of size n, Arthur uses O(log^6 n) random bits and Merlin uses O(log^2 n) nondeterministic bits. We derandomize this protocol for the case of solvable groups showing the following two results: (a) We give a uniform NP machine for solvable Group Non-Isomorphism, that works correctly on all but 2^polylog(n) inputs of any length n. Furthermore, this NP machine is always correct when the input groups are nonisomorphic. The NP machine is obtained by an unconditional derandomization of the AM protocol. (b) Under the assumption that EXP ⊈ i.o.PSPACE we get a complete derandomization of the above AM protocol. Thus, EXP ⊈ i.o.PSPACE implies that Group Isomorphism for solvable groups is in NP ∩ coNP.