The complexity of equivalence and isomorphism of systems of equations over finite groups

Gustav Nordh
2005 Theoretical Computer Science  
We study the computational complexity of the isomorphism and equivalence problems on systems of equations over a fixed finite group. We show that the equivalence problem is in P if the group is Abelian, and coNP-complete if the group is non-Abelian. We prove that if the group is non-Abelian, then the problem of deciding whether two systems of equations over the group are isomorphic is coNP-hard. If the group is Abelian, then the isomorphism problem is GRAPH ISOMORPHISM-hard. Moreover, if we
more » ... se the restriction that all equations are of bounded length, then we prove that the isomorphism problem for systems of equations over finite Abelian groups is GRAPH ISOMORPHISMcomplete. Finally, we prove that the problem of counting the number of isomorphisms of systems of equations is no harder than deciding whether there exist any isomorphisms at all.
doi:10.1016/j.tcs.2005.07.018 fatcat:63za63ve5rhsrhidxnz7b4xwbq