Group Theory

The mathematical analysis of symmetry

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This page will be of most interest to university undergraduate students of Group Theory who wish to enhance their understanding by some 'hands-on' experience of calculating with small finite groups.

I have written a simple interactive computer program as a 'group calculator' to teach yourself group theory.   The program should complement a text book on the subject.  The notes on its use are extensive and contain a course of graded worked examples.

The program exists in two versions, depending on how the finite groups are generated.  The 'PermGroups' program generates using permutations, whilst 'WordGroups' creates groups from abstract words on two generators.   By experimenting with both versions and following the examples described in the comprehensive notes, you will become hands-on familiar with some finite groups and their main features.

Both versions of the program generate finite groups of small order, calculate products of elements, conjugacy classes, commutators, subgroups and their cosets.

There is the option to restrict study to this introductory level material or to include the more advanced topics of group character tables and representation matrices, including characters lifted and induced from a subgroup.   The programs together can therefore be matched either to a Year 1/Year2 course, or to a Year3/Year 4 course.

Both programs are interactive to a limited extent through Yes or No answers to questions printed to screen, though the presentation of output is crude.   I give no guarantees of performance, but undergraduate students may find them helpful to support self-study.  They are offered in this spirit.

Index

PART 1: Outline of what the two versions of the program do

PART 2 : Worked Examples using WordGroups. This Part contains quite a lot of explanation of how to use the program, and the examples increase in complexity to provide a structured programme for self study.

        2.1 Not a finite group :

        2.2 Cyclic groups, C5 and C12

        2.3 Abelian direct products, C4C3 and C3C3

        2.4 A symmetric group S3 = a dihedral group D3

        2.5 Another dihedral group, D4

        2.6 Comparison of D4 (dihedral) with Q (quaternion)

        2.7 An abelian semi-direct product C3 x| C4

        2.8 An alternating group, A4

        2.9 The symmetric group, S4

        2.10 The five groups of order 20.

PART 3 : Worked Examples using PermGroups

        3.1 A cyclic group, C5 (Z5)

        3.2 Abelian direct products, CmCn

        3.3 A symmetric group S3 = a dihedral group D3

        3.4 S4 and its subgroups A4 and D4

        3.5 A non-abelian direct product S3 C3.

        3.6 Symmetric group S5 and alternating group A5

Download the Notes on Worked Examples (pdf)

Download WordGroups program (Zipped exe file)

Download PermGroups program (Zipped exe file)

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