Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

For simple random walk on aN-vertex graph, the mean time to cover all vertices is at leastcN log(N), wherec>0 is an absolute constant. This is deduced from a more general result about stationary finite-state reversible Markov chains. Under weak conditions, the covering time for such processes is at leastc times the covering time for the corresponding i.i.d. process.

Lower bounds for covering times for reversible Markov chains and random walks on graphs

/pdf/handbook-of-magma-functions-5anhrdxxj3.pdf

Handbook of magma functions

We prove that cubulated hyperbolic groups are virtually special. The proof relies on results of Haglund and Wise which also imply that they are linear groups, and quasi-convex subgroups are separable. A consequence is that closed hyperbolic 3-manifolds have finite-sheeted Haken covers, which resolves the virtual Haken question of Waldhausen and Thurston's virtual fibering question. An appendix to this paper by Agol, Groves, and Manning proves a generalization of the main result of "Residual finiteness, QCERF and fillings of hyperbolic groups".

/pdf/the-virtual-haken-conjecture-413qupzxg3.pdf

The virtual Haken conjecture

We show that the Graph Isomorphism (GI) problem and the related problems of String Isomorphism (under group action) (SI) and Coset Intersection (CI) can be solved in quasipolynomial (exp (logn) O(1) � ) time. The best previous bound for GI was exp(O( √ nlogn)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp( e O( √ n)), where n is the size of the permutation domain (Babai, 1983). The algorithm builds on Luks’s SI framework and attacks the barrier configurations for Luks’s algorithm by group theoretic “local certificates” and combinatorial canonical partitioning techniques. We show that in a well-defined sense, Johnson graphs are the only obstructions to effective canonical partitioning.

Graph Isomorphism in Quasipolynomial Time

From the Publisher:
This study in combinatorial group theory introduces the concept of automatic groups. It contains a succinct introduction to the theory of regular languages, a discussion of related topics in combinatorial group theory, and the connections between automatic groups and geometry which motivated the development of this new theory. It is of interest to mathematicians and computer scientists and includes open problems that will dominate the research for years to come.

Word processing in groups

This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups

History of Computational Group Theory BACKGROUND MATERIALS Fundamentals Group Actions Series Presentation of Groups Presentation of Subgroups Abelian Group Presentations Representation Theory, Modules, Extension, Derivations, and Complements Field Theory REPRESENTING GROUPS ON A COMPUTER Representing Groups on Computers The Use of Random Methods in CGT Some Structural Calculators Computing with Homorphisms COMPUTATION IN FINITE PERMUTATION GROUPS The Calculation of Orbits and Stabilizers Testing for Alt (W) and Sym (W) Finding Block Systems Bases and Strong Generating Sets Homomorphisms from Permutation Groups Backtrack Searches Sylow Subgroups, P-cores, and the Solvable Radical Applications COSET ENUMERATION The Basic Procedure Strategies for Coset Enumeration Presentations of Subgroups Finding All Subgroups Finding All Subgroups Up to a Given Index Applications PRESENTATION OF GIVEN GROUPS Finding a Presentation of a Given Group Finding a Presentation of a Strong Generating Set The Sims 'Verify' Algorithm REPRESENTATIONS, COHOMOLOGY, AND CHARACTERS Computation in Finite Fields Elemetary Computational Linear Algebra Factorizing Polynomials Over Finite Fields Testing KG-Models for Irreducibility - The Meataxe Related Computations Cohomology Computing Character Tables Structural Investigation of Matrix Groups COMPUTATION WITH POLYCYCLIC GROUPS Polycyclic Presentations Examples of Polycyclic Groups Subgroups and Membership Testing Factor Groups and Homomorphisms Subgroup Series Orbit-Stabilizer Methods Complements and Extensions Intersections, Centralizers, and Normalizers Automorphism Groups The Structure of Finite Solvable Groups COMPUTING QUOTIENTS OF FINITELY PRESENTED GROUPS Finite Quotients and Automorphism Groups of Finite Groups Abelian Quotients Practical Computation of the HNF and SNF P-Quotients of Finitely-Presented Groups ADVANCED COMPUTATIONS IN FINITE GROUPS Some Useful Subgroups Computing Composition and Chief Series Applications of the Solvable Radical Method Computing the Subgroups of a Finite Group Appication - Enumerating Finite Unlabelled Structures LIBRARIES AND DATABASES Primitive Permutation Groups Transitive Permutation Groups Perfect Groups The Small Groups Library Crystallorgraphic Groups Other Databases REWRITING SYSTEMS Monoid Systems Rewriting Systems Rewriting Systems in Monoids and Groups Rewriting Systems for Polycyclic Groups Verifying Nilpotency Applications FINITE STATE AUTOMATA AND AUTOMATIC GROUPS Finite State Automata Automatic Groups The Algorithm to Compute Shortlex Automatic Structures Related Algorithms Applications

Handbook of Computational Group Theory

A practical method is described for deciding whether or not a finite-dimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalistaion of the Parker-Norton ‘Meataxe’ algorithm, but it does not depend for its efficiency on the field being small. The principal tools involved are the calculation of the nullspace and the characteristic polynomial of a matrix over a finite field, and the factorisation of the latter. Related algorithms to determine absolute irreducibility and module isomorphism for irreducibles are also described. Details of an implementation in the GAP system, together with some performance analyses are included.

/pdf/testing-modules-for-irreducibility-4e5wp9j82v.pdf

Testing modules for irreducibility

A graph having 27 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.

Derek F. Holt

Papers

Word processing in groups

The Maximal Subgroups of the Low-Dimensional Finite Classical Groups

Handbook of Computational Group Theory

Testing modules for irreducibility

A graph which is edge transitive but not arc transitive