Section: Course Materials | M1: Groups and Group Actions (2022-23)

Course Term: Hilary & Trinity

Course Lecture Information: 8 lectures each in HT and TT

Course 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 get familiarised with the axiomatic approach to group theory and learn how to argue formally and abstractly. They will be able to apply the First Isomorphism Theorem and work with many examples of groups and group actions from various parts of mathematics. With the help of the Counting Lemma (also called Burnside's Lemma) they will be able to solve a variety of otherwise intractable counting problems and thus learn to appreciate the power of groups.

Course Synopsis:

HT (8 lectures)

Axioms for a group and for an Abelian group. Examples including geometric symmetry groups, matrix groups (\(GL_{n}\), \(SL_{n}\), \(O_{n}\), \(% SO_{n}\), \(U_{n}\)), 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 of order 8 or less up to isomorphism (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 \(G\rightarrow \mathrm{Sym}(S)\) associated with an action of \(G\) on \(S\). Cayley's Theorem. Symmetry groups of the tetrahedron and cube.

Section outline