Solvable Group Isomorphism Is (Almost) in NP ∩ coNP

Abstract

The Group Isomorphism problem consists in deciding whether two input groupsG1andG2given by their multiplication tables are isomorphic. We first give a 2-round Arthur-Merlin protocol for the Group Nonisomorphism problem such that on input groups (G1,G2) of sizen, Arthur usesO(log6n) random bits and Merlin usesO(log2n) 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 Nonisomorphism, that works correctly on all but 2logO(1)(n) inputs of any lengthn. Furthermore, this NP machine is always correct when the input groups are nonisomorphic. The NP machine is obtained by an unconditional derandomization of the aforesaid AM protocol.(b) Under the assumption that EXP \not\subseteq i.o--PSPACE we get a complete derandomization of the aforesaid AM protocol. Thus, EXP \not\subseteq i.o--PSPACE implies that Group Isomorphism for solvable groups is in NP ∩ coNP.