On the Complexity of Some Problems on Groups Input as Multiplication Tables

The Cayley Group Membership problem (CGM) is to input a groupoid (binary algebra) G given as a multiplication table, a subset X of G, and an element t of G, and to determine whether t can be expressed as a product of elements of X. For general groupoids CGM is P-complete, and for associative algebras (semigroups) it is NL-complete. Here we investigate CGM for particular classes of groups. The problem for general groups is in SL (symmetric log space), but any kind of hardness result seems difficult because it would require constructing the entire multiplication table of a group. We introduce the complexity class FOLL, or FO(log logn), of problems solvable by uniform polysize circuit families of unbounded fan-in and depth O(log log n). Since parity is not in FOLL, no problem in FOLL can be complete for any class containing parity, such as NC1, L, or SL. But FOLL is not known to be contained even in SL. We show that CGM for cyclic groups is in FOLL\L and that CGM for abelian groups is in FOLL. We then partially extend our method to solvable groups, showing that CGM for groups of constant solvability class is in FOLL and that CGM for nilpotent groups can be solved by poly-size circuits of depth O((log logn)2). (Thus the latter problem is provably not complete for any class containing parity.) We also consider the problem of testing for various properties of a group input as a table: we prove that cyclicity and nilpotency can each be tested in FOLL \ L. Finally, we examine the implications of our results for the complexity of iterated multiplication, powering, and division of integers, in the context of the recent results of Chiu, Davida, and Litow. Supported by the Québec FCAR (McKenzie), by the NSERC of Canada (McKenzie), and by the German DFG who kindly made possible McKenzie’s sabbatical leave at Universität Tübingen. Emails: [email protected], fkadau,[email protected], [email protected]