Studies in Structure

Frontmatter

1. Modular Arithmetic

Abstract

The arithmetic of odd and even numbers is well-known. If an odd number is combined with an even number by addition, the result is odd; if combined by multiplication the result is even. Tables can be constructed for all possible combinations:

Joan M. Holland

2. GROUPS OF ORDER SIX: formal definition of a group

Abstract

What rotations can we perform on a regular hexagon subject to the condition that afterwards it is to occupy the same hexagonal space?

Joan M. Holland

3. Some Ways of Organising a Group

Abstract

The number a³ is often defined as the product of three factors each equal to a. Another way of defining it is to say that a³ is the result of multiplying the identity element 1 by a three times in succession. This has the advantage of making sense of the rather baffling expression a⁰. To multiply 1 by a zero times naturally leaves it unchanged suggesting that a⁰ = 1. It is convenient to adopt this notation in the study of groups and to describe any cyclic group of order n as consisting of the elements 1, a, a², a³ ... a^{n − 1}. For example the non-zero residue classes (mod 7), may be written 3⁰, 3¹, 3², 3³, 3⁴, 3⁵ or otherwise 1, 3, 2, 6, 4, 5. The multiplication table of the elements written in this order shows the characteristic diagonal pattern of a cyclic group.

Joan M. Holland

4. Fibonacci Sequences

Abstract

Many readers will be familiar with the famous sequence

in which each term after the second is obtained by adding the previous two terms. It was known at least as early as 1226 when Leonardo Fibonacci of Pisa propounded his problem concerning the proliferation of a hypothetical pair of rabbits: each pair of newborn rabbits is assumed to bear its first pair two months later and thereafter to bear one pair a month. (See N. N. Vorob’ev, Fibonacci Numbers, Popular Lectures in Mathematics, Vol. 2).

Joan M. Holland

5. Biological Illustrations of Fibonacci Sequences

Abstract

Many interesting accounts have been written of the ways in which Fibonacci numbers keep cropping up in biological situations. We began the last chapter with the development of a rabbit colony, admittedly rather an artificial example. The ancestry of a male bee has been described in Professor F.W. Land, The Language of Mathematics. Here is the drone’s family tree, taking into account that he comes from an unfertilised egg of a queen bee: Successive generations of ancestors follow the Fibonacci sequence; this assumes, however, that all queens in the ancestry are distinct, which is not necessarily true.

Joan M. Holland

6. New Groups from Old and Groups within Groups

Abstract

This chapter has two main themes and one subsidiary, all inter-related. Section 1–7 deal with direct product groups, sections 8–12 with quotient groups, and section 13 with equivalence classes. Quotient groups were first mentioned on an experimental basis in section 3.4. Those introductory ideas are again used in the first portion of this chapter. A more detailed and rigorous account of quotient groups follows in the second portion. The reader who prefers not to postpone the rigorous approach could read sections 8–12 before sections 1–7. The final section of the chapter is an elementary treatment of equivalence, for the benefit of those readers who may not have previously met this important concept in its general algebraic sense; it underlies the discussion of modular arithmetic in section 2.1 and is essential to the understanding of the rôle of cosets in quotient groups which is developed in the next chapter.

Joan M. Holland

7. Fields, Rings, and Homomorphisms: Illustrations from the Fibonacci Sequence

Abstract

It becomes necessary to explain the technical term ‘field’ in order to be able to use it with precision. A field is a structure consisting of a set whose elements are related by two binary operations (i.e. rules of combination) as opposed to a group whose elements are related by one only. The elements are closed under both operations, usually addition and multiplication, and each operation has the commutative and associative properties. In symbols:

Joan M. Holland

8. Polynomials and Finite Fields

Abstract

In section 7.1 a field was defined and examples were mentioned which included the finite field Z_p obtained by forming the residue classes for a prime modulus p. In this chapter we shall consider other finite fields obtained by combining Z_p with powers of x to form polynomial expressions in x: but first it is advisable to extend the ideas on congruency and on the solution of congruencies which were introduced in Chapter 1. Section 1.10 is particularly relevant.

Joan M. Holland

9. Mappings, Permutation Groups, and Groups of Automorphisms

Abstract

We have already used the term ‘mapping’ in several places without defining it precisely. It is appropriate now to formalise the idea so that we may go on to study permutations which form a particular class of mapping and also to explore further the idea of automorphism.

Joan M. Holland

10. Three Substitution Groups. Illustrations of the Symmetric Group S4

Abstract

In section 8.5 we constructed a field G.F.2³ consisting of eight polynomials of degree 2 over the field Z₂ using an indeterminate R which satisfied the irreducible cubic equation R³ = R + 1. The seven non-zero elements form a cyclic group under multiplication. This enables us to set out a table of reciprocals among its elements as follows:

Joan M. Holland

11. Three Examples of Groups Associated with Repetitive Patterns

Abstract

Consider the cyclic pattern below. It suggests the finite group C₆ consisting of six rotations clockwise of 0°, 60°, 120°, …: any of these applied to the pattern leaves it invariant. The set of transformations might be named 1, T, T², T³, T⁴, T⁵ or in a slightly different form T⁻², T⁻¹, 1, T, T², T³: then T stands for a clockwise rotation of 60°, T³ for the sanie rotation repeated three times and T⁻¹ for the inverse rotation of 60° anti-clockwise.

Joan M. Holland

12. What is a Group?

Abstract

I remember as a child being highly amused by an anecdote about a traveller in Ireland: being completely lost he enquired of a passer-by how to find his way to Ballymoy. He received first a dismayed look and then the reply ‘Ah now, if you want to go to Ballymoy you shouldn’t start from here!’

Joan M. Holland

Backmatter