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

Polynomial-time normalizers

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and ComplexityFor an integer constant d \textgreater 0, let Gamma(d) denote the class of finite groups all of whose nonabelian composition factors lie in S-d; in particular, Gamma(d) includes all solvable groups. Motivated by applications to graph-isomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, the problems of finding set stabilizers, intersections and centralizers have all been shown to be polynomial-time computable. A notable open issue for the class Gamma(d) has been the question of whether normalizers can be found in polynomial time. We resolve this question in the affirmative. We prove that, given permutation groups G, H \textless= Sym(Omega) such that G is an element of Gamma(d), the normalizer of H in G can be found in polynomial time. Among other new procedures, our method includes a key subroutine to solve the problem of finding stabilizers of subspaces in linear representations of permutation groups in Gamma(d)

Polynomial-time normalizers

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexit