Synopsis for Groups and Group Actions

Overview

Abstract algebra evolved in the twentieth century out of nineteenth century discoveries in algebra, number theory and geometry. It is a highly developed example of the power of generalisation and axiomatisation in mathematics. The group is an important first example of an abstract, algebraic structure and groups permeate much of mathematics particularly where there is an aspect of symmetry involved. Moving on from examples and the theory of groups, we will also see how groups act on sets (e.g. permutations on sets, matrix groups on vectors) and apply these results to several geometric examples and more widely.

Learning Outcomes

Students will appreciate the value of abstraction and meet many examples of groups and group actions from around mathematics. Beyond theoretic aspects of group theory students will also see the value of these methods in the generality of the approach and also to otherwise intractable counting problems.

Synopsis

HT (8 lectures)

Axioms for a group and for an Abelian group. Examples including geometric symmetry groups, matrix groups (, , , , ), cyclic groups. Products of groups.

Permutations of a finite set under composition. Cycles and cycle notation. Order. Transpositions; every permutation may be expressed as a product of transpositions. The parity of a permutation is well-defined via determinants. Conjugacy in permutation groups.

Subgroups; examples. Intersections. The subgroup generated by a subset of a group. A subgroup of a cyclic group is cyclic. Connection with hcf and lcm. Bezout's Lemma.

Recap on equivalence relations including congruence mod n and conjugacy in a group. Proof that equivalence classes partition a set. Cosets and Lagrange's Theorem; examples. The order of an element. Fermat's Little Theorem.

TT (8 Lectures)

Isomorphisms, examples. Groups up to isomorphism of order 8 (stated without proof). Homomorphisms of groups with motivating examples. Kernels. Images. Normal subgroups. Quotient groups; examples. First Isomorphism Theorem. Simple examples determining all homomorphisms between groups.

Group actions; examples. Definition of orbits and stabilizers. Transitivity. Orbits partition the set. Stabilizers are subgroups.

Orbit-stabilizer Theorem. Examples and applications including Cauchy's Theorem and to conjugacy classes.

Orbit-counting formula. Examples.

The representation associated with an action of on . Cayley's Theorem. Symmetry groups of the tetrahedron and cube.

Reading

1. M. A. Armstrong Groups and Symmetry (Springer, 1997)

Alternative Reading

1. R. B. J. T. Allenby Rings, Fields and Groups (Second Edition, Edward Arnold, 1999) [Out of print, but available in many libraries also via Amazon.]
2. Peter J. Cameron, Introduction to Algebra, (Oxford University Press, 1998).
3. John B. Fraleigh, A First Course in Abstract Algebra (Seventh edition, Pearson, 2002).
4. W. Keith Nicholson, Introduction to Abstract Algebra (Second edition, John Wiley, 1999).
5. Joseph J. Rotman, A First Course in Abstract Algebra (Second edition, Prentice-Hall, 2000).
6. Joseph Gallian Contemporary Abstract Algebra (Houghton Mifflin, 2006)
   ISBN 0618122141.