Algebra II

MATH2581

Lecturer : Herbert Gangl

Term : Epiphany 2014

Lectures :
  • Tuesday 4:00pm in D110
  • Thursday 10:00am in W103
Problem Classes:
  • Thursday 3:00pm in CG93 (even weeks only)

Tutorials

  • In odd weeks.
Reading suggestions:

    Any of the following four references covers most of the material in the lectures (in rather different form).
    Examples and motivation are taken from various sources.

  • Peter Cameron: Introduction to algebra, Oxford University Press.

  • R.B.J.T. Allenby: Rings, fields and groups: an introduction to abstract algebra, Arnold.

  • M.A. Armstrong: Groups and symmetry, Springer.

  • There are also rather detailed online notes by Tony Gaglione which cover similar material.

      For more basic and detailed coverage you may want to consider

  • Cameron very nice (free) online Notes on Algebraic Structures (in particular Chapter 3).

  • An even more basic and detailed reference is perhaps Chapter 6 of C. Whitehead: Guide to Abstract Algebra. MacMillan.

Lecture notes:

    Here are scanned notes from all the lectures, taken by Yingxue Shang. Moreover, there is a new and somewhat re-structured version of the lecture notes as well as the older version for the course so far. And here is a link to the notes for a previous Epiphany term on Algebra.

    (a)Musical notes:

    Poem: There is also a songified version (Safari plays it directly, Firefox and Google Chrome allow to download it) of the latest poem from the lectures.

    Pure Maths joy: A wonderful a cappella treatment of the finite simple group of order 2, performed by The Klein Four.

    A Capella Science: A simply amazing a capella rendering of the famous Queen song, called Bohemian Gravity, performed by Tim Blais.

Assignments

Homework Date Hand in Solutions
Homework Set 1 pdf 21.1. 28.1. pdf
Homework Set 2 pdf 30.1. 6.2. pdf
Homework Set 3 pdf 4.2. 13.2. pdf
Homework Set 4 pdf 11.2. 20.2. pdf
Homework Set 5 pdf 18.2. 27.2. pdf
Homework Set 6 pdf 27.2. 6.3. pdf
Homework Set 7 pdf 6.3. 13.3. pdf
Homework Set 8 pdf 13.3. 20.3. Q4,5 Sheet 17(!) pdf


Lecture Outline

Date Outline
20.3. Fundamental Theorem of finitely generated abelian groups; how many elements of a given order?
18.3. Towards classification of finitely generated abelian groups.
13.3. Classification of groups of order p^2.
11.3. Conjugacy classes in A_n, normal subgroups in S_4 and A_4.
6.3. Conjugacy classes and normal subgroups of S_n and A_n.
4.3. Proof of Cauchy's Theorem, Classification of groups of order 2p.
27.2. Orbit-Stabiliser Theorem, Cauchy's Theorem.
25.2. Conjugacy as a group action, Orbit-Stabiliser Theorem.
20.2. Group actions, orbits and stabilisers.
18.2. Proof of Cayley's Theorem. Group actions.
13.2. Cayley's Theorem; rotations of the cube as permutation groups.
11.2. Distinguishing and identifying groups; direct product, isomorphisms.
6.2. Odd/even permutations, the alternating groups.
4.2. Permutations, symmetric groups, cycle notation.
30.1. Direct product, isomorphism (recap), permutations, towards symmetric groups.
28.1. Conjugacy classes, the centre of a group.
23.1. Cosets, normal subgroups, factor groups. Conjugacy classes.
21.1. Overview, and quick recap of basic notions surrounding the concept of a group
(subgroups, generators, orders, Lagrange's theorem, homomorphisms, normal subgroups, factor groups).
Examples.

Last modified: 14.3.2014.