The complexity of solving equations over finite groups

  • Authors:

  • Mikael Goldmann;Alexander Russell

  • Affiliations:

  • Numerical Analysis and Computer Science, Royal Insitute of Technology, Stockholm, Sweden;Department of Computer Science and Engineering, University of Connecticut, Storrs, Connecticut

  • Venue:
  • Information and Computation
  • Year:
  • 2002

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Abstract

We study the computational complexity of solving systems of equations over a finite group. An equation over a group G is an expression of the form w1.w2....wk = 1G, where each wi is either a variable, an inverted variable, or a group constant and 1G is the identity element of G . A solution to such an equation is an assignment of the variables (to values in G) which realizes the equality. A system of equations is a collection of such equations; a solution is then an assigmnent which simultaneously realizes each equation. We show that the problem of determining if a (single) equation has a solution is NP-complete for all nonsolvable groups G. For nilpotent groups, this same problem is shown to be in P. The analogous problem for systems of such equations is shown to be NP-complete if G is non-Abelian, and in P otherwise. Finally, we observe some connections between these problems and the theory of nonuniform automata.