GAP in Groups and Semigroups

Recommended Prerequisites

Knowledge of group theory obtained in a first undergraduate course of Algebra.   

Teaching Methods

The teaching/learning process will follow an interactive approach in which students will immediately apply the principles taught through experimentation with GAP on individual assignments. The results will be discussed immediately.

Learning Outcomes

This learning unit (LU) aims at providing the fundamental knowledge and competencies regarding the principles, concepts, models and techniques that constitute the area of computational algebra as applied to group and semigroup theory, namely, the fundamentals of group and semigroup theory; modelling of algebraic problems algorithmically; applications of high performance computing to resolve open problems; implementing of software packages.

After completion of this LU, students should be able to:

- Recognise the importance of computational algebra in contemporary abstract algebra, both in its successes and limitations;

- Identify, classify and integrate the principles, main models, algorithms and techniques of computational algebra;

- Identify, analyze, categorize and evaluate existing implementations; develop new software to solve problems in semigroup theory.

- program in the computational algebra system Groups, Algorithms, and Programming (GAP – www.gapsystem.org).   

Work Placement(s)

No

Syllabus

1) The foundations of semigroup and group theory; history and development of the computational algebra; an introduction to the GAP system; algorithms and implementations; using libraries and conducting searches; finitely presented groups and semigroups; permutation groups and transformation semigroups; programming in GAP.

2) Areas of algebra in which computational algebra has been successfully applied.

3) Using GAP to solve problems in abstract algebra; major computational algebra systems and libraries; modelling of algebraic problems algorithmically; high performance computing; applications to resolve open problems; implementing of software packages; main references for further study.   

Assessment Methods

Assessment
Exam: 30.0%
Resolution Problems: 35.0%
Research work: 35.0%

Bibliography

- “Fundamentals of Semigroup Theory”, J. M. Howie, Oxford University Press, 1995.

- “Handbook of computational group theory”, D. Holt with B. Eick and E. O'Brien, CRC Press, 2004.

- “Permutation groups”, J. D. Dixon and B. Mortimer, Springer-Verlag, 1996.

- “Inverse Semigroups”, M. V. Lawson, World Scientific, 1998.

- The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.4.12; 2008. (http://www.gap-