Derivation towers of Lie algebras

AbstractFor each natural number n there exist finite dimensional centerless Lie algebras, L, whose derivation towers L ◁ Der(L) ◁ Der(Der(L)) ◁ … do not stabilize in less than n steps

Parallel algorithms for solvable permutation groups

AbstractA number of basic problems involving solvable and nilpotent permutation groups are shown to have fast parallel solutions. Testing solvability is in NC as well as, for solvable groups, finding order, testing membership, finding centralizers, finding centers, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in NC, find a central composition series, and find pointwise stabilizers of sets. The latter is applied to an instance of graph isomorphism. A useful tool is the observation that the problem of finding the smallest subspace containing a given set of vectors and closed under a given set of linear transformations (all over a small field) belongs to NC

Computing in quotient groups

We present polynomial-time algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomial-time for permutation groups. Since it is not computationally feasible to represent G/K itself as a permutation group, the methodology for the quotient-group versions of such problems frequently differ markedly from the procedures that have been observed for the K = 1 subcases. Whereas the algorithms for permutation groups may have rested on elementary notions, procedures underlying the extension to quotient groups often utilize deep knowledge of the structure of the group. In some instances, we present algorithms for problems that were not previously known to be in polynomial time, even for permutation groups themselves (K = 1). These problems apparently required access to quotients. 1

Testing isomorphism of modules

We present a new deterministic algorithm to test constructively for isomorphism between two given finite-dimensional modules of a finitely generated algebra. The algorithm uses only basic field operations; for arbitrary fields, this is not possible with the existing methodology. Furthermore, the number of field operations used by the algorithm is bounded by a polynomial in the length of the input. The algorithm has been implemented in the computer algebra system Magma and we report on its performance. Our approach has applications to other problems concerning decompositions of modules.