University of Wales, Bangor - School of Informatics Mathematics H2 Modules 2000/2001

Lecturer: Prof T Porter

Recommended for further reading:

• Modern Algebra 3rd edition, J R Durbin, Wiley, 1992, ISBN 0471559121, 17.95, (QA162.D87).
• A First Course in Abstract Algebra, J. Fraleigh, Addison Wesley, Reading, Mass. 1992.
• Knots and Surfaces, N.D.Gilbert & T.Porter, OUP 1994 (QA612.2.G55) Paperback edition.
• (Highly recommended for basic revision of Group Theory from H1)
  Mathematical Groups, Teach Yourself Books, T.Barnard & H.Neill, Hodder & Stoughton, ISBN 0-340-67012-6, 6.99.

Aims:

• To introduce students to concepts of elementary group theory through the analysis of presentations of groups.
• To help in the development of techniques of rewriting and manipulation in algebra.
• To encourage a systematic approach in the manipulation of symbolic representations.
• To initiate students in the careful use of the algorithms of combinatorial group theory.

Objectives
At the end of the course students should be able to:

• Apply the Todd-Coxeter heuristic to a finitely presented group so as to obtain the index of a subgroup, a permutation representation and a Cayley diagram for the group.
• Use presentations to determine the automorphisms of a group, at least for elementary examples.
• Handle, with confidence, questions relating to conjugacy, normal subgroups and quotient groups within a range of simple examples.
• Use Tietze transformations to simplify presentations and the techniques of Fox derivatives and the Alexander matrix to analyse if two presentations present the same group.
• Prove, and apply with confidence, elementary results from Group Theory (Isomorphism Theorems, and the elementary results on free groups).

Summary:
Groups occur naturally as symmetries of objects, e.g. the dihedral group D_3 is the symmetry group of the triangle. Often a group is specified in terms of various generating elements and relations between them, e.g. if R denotes a rotation through 120 degrees and S a flip over, then D_3 is generated by R and S and we know R^3=1, S^2=1 and (RS)^2=1. Does this information determine D_3 completely? Can one see the symmetry aspect of a group if we are given a presentation of it?
Given a presentation of G, how can one answer questions about G, such as
"What are the automorphisms of G?", "What subgroups does G have?".
The course examines group theory through presentations.

Prerequisites: G2M31.

Syllabus:

Review of Group Theory (from G2M31):
Definitions of: group homomorphism; order of an element; abelian group; kernel and image; commutator; quotient group. Statement and proof of the first Isomorphism Theorem and Lagrange's Theorem

Introduction of Standard Examples:
C_n, S_3, S_n, D_n; triangle groups D(l,m,n); groups of matrices, e.g. GL(2,3). Idea of a representation (both matrix and permutation). Cayley's Theorem.

Informal introduction to presentations:
Working with presentations: C_n, D_3, D_n, etc. (Elementary use of GAP on handout.) Conjugacy, normal subgroups, center, etc.

Van Dyck's Theorem and the Substitution Test:
Use to list endomorphisms, epimorphisms, and to pick out automorphisms in simple examples. Detailed analysis of C_n, D_3, and D_4. Exercises on D_n.

Free Groups and formal results on presentations:
Equivalent presentations, Tietze transformations and Tietze's Theorem. Examples of use.

The Todd-Coxeter Algorithm:
Simple examples by hand. Cayley graph and permutations representations from the output. (Harder examples using GAP.)

Cayley graph of a presentation:
Automorphisms of a graph; Frucht's Theorem.

Assessment:
2 hour end of semester closed book examination 100%

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