Computation with Finitely Presented Groups

This book describes the basic algorithmic ideas behind accepted methods for computing with finitely presented groups.

Charles C. Sims (Author)

9780521432139, Cambridge University Press

Hardback, published 28 January 1994

624 pages
24.1 x 16 x 3.3 cm, 0.975 kg

"this book is a very interesting treatment of the computational aspects of combinatorial group theory. It is well-written, nicely illustrating the algorithms presented with many examples. Also, some remarks on the history of the field are included. In adition, many exercises are provided throughout...this is a very valuable book that is well-suited as a textbook for a graduate course on computational group theory. It addresses students of mahtematics and of computer science alike, providing the necessary background for both. In addition, this book will be of good use as a reference source for computational aspects of combinatorial group theory." Friedrich Otto, Mathematical Reviews

Research in computational group theory, an active subfield of computational algebra, has emphasised three areas: finite permutation groups, finite solvable groups, and finitely presented groups. This book deals with the third of these areas. The author emphasises the connections with fundamental algorithms from theoretical computer science, particularly the theory of automata and formal languages, computational number theory, and computational commutative algebra. The LLL lattice reduction algorithm and various algorithms for Hermite and Smith normal forms from computational number theory are used to study the abelian quotients of a finitely presented group. The work of Baumslag, Cannonito and Miller on computing nonabelian polycyclic quotients is described as a generalisation of Buchberger's Gröbner basis methods to right ideals in the integral group ring of a polycyclic group. Researchers in computational group theory, mathematicians interested in finitely presented groups and theoretical computer scientists will find this book useful.

1. Basic concepts
2. Rewriting systems
3. Automata and rational languages
4. Subgroups of free products of cyclic groups
5. Coset enumeration
6. The Reidemeister-Schreier procedure
7. Generalized automata
8. Abelian groups
9. Polycyclic groups
10. Module bases
11. Quotient groups.

Subject Areas: Mathematical theory of computation [UYA], Groups & group theory [PBG]