Introduction to Algebra

This page contains supporting material for the textbook Introduction to Algebra, by Peter J. Cameron, published by Oxford University Press in April 1998.

The ISBNs for the book are

019 850195 1 (hardback)
019 850194 3 (paperback)

This site will contain solutions to the exercises, further material, problems, corrections, and links to other sites of interest to algebraists.

From the reviews:

By Plamen Koshlukov in Zentralblatt für Mathematik:

... a very good textbook in Algebra... The lecturers who are going to decide to use this book for their classes will make a good choice; they and their students will benefit a lot from this book.

A simple argument

Here is a much shorter proof that the commutativity of addition follows from the other axioms in a ring with identity (Exercise 2.7).

Expand (1+1)(x+y) in two different ways:

(1+1)(x+y) = 1(x+y)+1(x+y) = x+y+x+y
(1+1)(x+y) = (1+1)x+(1+1)y = x+x+y+y

Now cancellation gives y+x = x+y.

Solutions to exercises

The PDF files below give the solutions to all the exercises in Chapters 1-3. Others will be added.

Misprints

A list of corrections is available. Please send me reports of misprints (an email link is given below).

A problem

Prove that there is no simple group of order 3³.7.13.409.

Miscellany

1. Here is a picture of Muhammad Al-Khwarizmi, who gave us the word "Algebra".

2. Can you explain what the following is about?

Curiously enough, the twelve-tone system has no zero in it. Given a series: 3, 5, 2, 7, 10, 8, 11, 9, 1, 6, 4, 12 and the plan of obtaining its inversion by numbers which when added to the corresponding ones of the original series will give 12, one obtains 9, 7, 10, 5, 2, 4, 1, 3, 11, 6, 8 and 12. For in this system 12 plus 12 equals 12. There is not enough of zero in it.

John Cage, "Eric Satie", in Silence: Lectures and Writings, Calder and Boyars, 1968.

Read my solution

3. Hermann Weyl on Proposition 2.12(c), the statement that (ST)^-1=T^-1S^-1:

With this rule, although perhaps not with its mathematical expression, you are all familiar. When you dress, it is not immaterial in which order you perform the operations; and when in dressing you start with the shirt and end up with the coat, then in undressing you observe the opposite order; first take off the coat and the shirt comes last.

Hermann Weyl, Symmetry, Princeton University Press, 1952.

Links

Peter Cameron's homepage at Queen Mary, University of London
Group-Pub-Forum homepage at the University of Bath
Permutation groups resources at Queen Mary, University of London
ATLAS of Finite Group Representations at the University of Birmingham
GAP homepage at St Andrews University
Overview of MAGMA (mirror site at the University of Bath - original at the University of Sydney)
History of Matrices and Determinants and History of Group Theory from the MacTutor History of Mathematics Archive at St Andrews University
A Survey of Venn diagrams by Frank Ruskey, from the Electronic Journal of Combinatorics
A Brief History of Algebra and Computing: An Eclectic Oxonian View by Jonathan P. Bowen at the University of Reading
The Art of Algebra from Al-Khwarizmi to Viete by Karen Hunger Parshall at the University of Virginia
Solving the quintic (worked example in Mathematica)
Semigroups directory at the University of Southampton
Nearrings homepage at the University of Linz
Theory and Applications of Categories (electronic journal)

The mailing list group-pub-forum is for the discussion of any aspect of Group Theory. The reason for the name is that the spirit is supposed to be that of a conversation in a pub at a Group Theory conference.

GAP and MAGMA are systems for computation with algebraic structures including groups and rings.

Peter J. Cameron
4 January 2001.