RSS Feed for the unit Symmetry

Introduction

In this unit we use the geometric concept of symmetry to introduce some of the basic ideas of group theory, including group tables, and the four properties, or axioms, that define a group.

Please note that this unit is presented through a series of PDF documents.

Learning Outcomes

By the end of this unit you should be able to:

explain what is meant by a symmetry of a plane figure;
specify symmetries of a bounded plane figure as rotations or reflections;
describe some properties of the set of symmetries of a plane figure;
explain the difference between direct and indirect symmetries;
use a two-line symbol to represent a symmetry;
describe geometrically the symmetry of a given figure which corresponds to a given two-line symbol;
find the composite of two symmetries given as two-line symbols;
find the inverse of a symmetry given as a two-line symbol;
write down a Cayley table for the set of symmetries of a plane figure;
appreciate how certain properties of the set of symmetries of a figure feature in a Cayley table;
explain the meaning of the terms group, Abelian group and the order of a group;
give examples of finite groups and infinite groups;
determine whether a given set and binary operation form a group by checking the group axioms;
deduce information from a given Cayley table;
understand that the identity in a group is unique;
understand that each element in a group has a unique inverse;
recognise how the uniqueness properties can be proved from the group axioms;
explain the connections between properties of a group table and the group axioms;
describe the symmetries of some bounded three-dimensional figures;
use two-line symbols to denote symmetries of three-dimensional figures, and to form composites and inverses of such symmetries;
count the number of symmetries of certain polyhedra;
understand why there are exactly five regular polyhedra.

1 Symmetry in two dimensions

In Section 1 we discuss intuitive ideas of symmetry for a two-dimensional figure, and define the set of symmetries of such a figure. We then view these symmetries as functions that combine under composition, and show that the resulting structure has properties known as closure, identity, inverses and associativity. We use these properties to define a group in Section 3.

Click 'View document' below to open Section 1 (15 pages, 623KB).

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2 Representing symmetries

In Section 2 we develop an algebraic notation for recording symmetries, and demonstrate how to use the notation to calculate composites of symmetries and the inverse of a symmetry.

Click 'View document' below to open Section 2 (9 pages, 504KB).

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3 Group axioms

Section 3 is an audio section. We begin by defining the terms group, Abelian group and order of a group. We then demonstrate how to check the group axioms, and we extend the examples of groups that we use to include groups of numbers – the modular arithmetics, the integers and the real numbers.

Click 'View document' below to open Section 3 (11 pages, 703KB).

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Please listen to the audio clips below when you are instructed to do so.

Click play to listen to the audio clip for frames 1 to 2 (8 minutes).

Listen in separate player Click play to start.

Click play to listen to the audio clip for frames 3 to 6 (7 minutes).

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Click play to listen to the audio clip for frames 7 to 8 (4 minutes).

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Click play to listen to the audio clip for frames 9 to 10 (3 minutes).

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Click play to listen to the audio clip for frames 11 to 13 (5 minutes).

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Click play to listen to the audio clip for frames 14 to 15 (5 minutes).

Listen in separate player Click play to start.

4 Proofs in group theory

In Section 4 we prove that some of the properties of the groups appearing earlier in the unit are, in fact, general properties shared by all groups. In particular, we prove that in any group the identity element is unique, and that each element has a unique inverse.

Click 'View document' below to open Section 4 (9 pages, 237KB).

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