Ákos Seress

Permutation groups in NC

We show that the basic problems of permutation group manipulation admit efficient parallel solutions. Given a permutation group G by a list of generators, we find a set of NC-efficient strong generators in NC. Using this, we show, that the following problems are in NC: membership in G; determining the order of G; finding the center of G; finding a composition series of G along with permutation representations of each composition factor. Moreover, given G, we are able to find the pointwise stabilizer of a set in NC. One consequence is that isomorphism of graphs with bounded multiplicity of eigenvalues is in NC. The analysis of the algorithms depends, in several ways, on consequences of the classification of finite simple groups.

Black box classical groups

Introduction Preliminaries Special linear groups:

On the diameter of Cayley graphs of the symmetric group

\mathrm {PSL} (d,q)

Fast management of permutation groups

Orthogonal groups:

Primitive Groups with no Regular Orbits on the Set of Subsets

\mathrm{P}\Omega^\varepsilon(d,q)

A black-box group algorithm for recognizing finite symmetric and alternating groups, I

Symplectic groups:

Nearly linear time algorithms for permutation groups with a small base

\mathrm{PSp}(2m,q)

Graphs of prescribed girth and bi-degree

Unitary groups:

Polynomial-time theory of matrix groups

\mathrm{PSU}(d,q)

On vertex-transitive, non-Cayley graphs of order pqr

Proofs of Theorems 1.1 and 1.1, and of corollaries 1.2-1.4 Permutation group algorithms Concluding remarks References.