Lectures:

• Tuesday 1:15 in CLC013 
• Wednesday 10:00 in CLC013
  
• Tutorial: weekly in various groups
  

Office hours in Easter Term:  Wednesday, 13/5 11-12. Thursday, 14/5 11-12, Monday, 18/5 11-12, Thursday, 21/5 10-11 or by appointment

Syllabus: In this course we study sets within which there are ideas of addition and/or multiplication. We start by looking at the set of integers and, in particular, the consequences which flow from the simple observation that we can divide one integer by another to obtain an integer quotient and a remainder
which is less than the divisor. On the one hand, this leads to a quick method for finding greatest common divisors and to a discussion of primes and factorisation. On the other, we discover modulo arithmetic, where we are allowed to “forget” multiples of a given fixed integer. (So, although “2+2 = 5” is still not a very useful idea, “7+7 = 2” has its moments — hours, to be more precise!). We go on to discuss examples of groups and some of the properties that they share. Groups are sets which (amongst other things) have a method (which may be thought of as addition or multiplication) of combining two elements to make another. The following are examples of groups: the set of integers (with the method of addition), the nonzero real numbers (multiplication) and invertible 2 by 2 matrices (multiplication). The permutations of a set and the symmetries of regular polygons give further examples. While the examples used will be mainly concrete rather than abstract, we shall be learning about the importance of correct algebraic notation and developing tools (theorems and ideas) to help us
determine whether the differences between the groups that we discover are fundamental or only superficial.

Topics:

• Formal definition of group, examples.
  
• Prime numbers, infinite number of primes, gcd’s, extended Euclidean algorithm in Z, Fundamental Theorem of Arithmetic. Application: rational roots of elements in Q[X].
  
• Congruences, residues, Z/nZ as a ring, (Z/nZ)* as set of elements in Z/nZ with  ab =1 for some b in Z/nZ. Solving simple congruences.
  
• Euler Phi–function.  Euler-Fermat ‘by hand’. Discuss public key cryptography.
• Permutations: cycle representation (also inverses, composition, conjugation), sign of permutation.
• Groups revisited. Notion of subgroup (with examples), order of an element, of a group, cosets Lagrange’s Theorem. Group homomorphism, isomorphism, equivalence of Z/mnZ  and Z/mZ x Z/nZ if gcd (m,n) = 1.
• Examples of groups defined by symmetries of sets. Dihedral groups, alternating groups.

Notes:  Class notes are here. A slightly different version is this. (The changes are minimal, however, it does include the dihedral group at the very end). While I will follow the outline of these notes, I might deviate on occasion!

Assignments:  Set each Wednesday; to be handed in to the tutor by the following week; returned on the follow-up tutorial.

Problems:

 Problem Sheet   
• HW1: Hand in week 12:  1(a),1(b),3,4(a). Prepare for tutorials: 1(c),2,5,4(b)
  
• HW2: Hand in week 13: 6,7,13. Prepare for tutorials: 9,12,14, 15,16
• HW3: Hand in week 14: 18 (b),(e),(f), 21 (d),(e),(f). Prepare for tutorials: 18(c), 21(b),(c), 19, 20
• HW4: Hand in week 15: 22(c),24(b),28. Prepare for tutorials: 22(b),23,24(c),27(a)
• HW5: Hand in week 16: 25(a),26(a),27(d),30(b). Prepare for tutorials: 25(b),26(b),30(d-f), 29 (as far as you want)
• HW6: Hand in week 17: 31(a),32 (corresponding to 31(a)),33(c),38. Prepare for tutorials: 31(b). 32(corresponding to 31(b)),33(d),39
• HW7: Hand in week 18: 36,43,49. Prepare for tutorials: 37,42,44
• HW8: (Virtual) hand in week 19: 48,50,52. Prepare for tutorials: 45,47,51,53
• Problems for Easter vacation: 46,53 (i,j),54 (a-c). Determine the symmetric group of a nonsquare rectangle.
  

Hints and Solutions to the Problems:

 Solutions   

Additional Course Notes:

  Concerning the first few lectures (pdf)      

  Further Online Notes on the Chinese Remainder Theorem (pdf)        

  Notes on basics about groups by Jonathan Arnold

  Link to a survey article on (twin) primes  by D.A. Goldston (more advanced, but first few pages perhaps understandable)

Recommended books:

• Peter Cameron: Introduction to Algebra; Oxford Science Publications
• Carol Whitehead: Guide to Abstract Algebra; Palgrave Mathematical Guides
• R.B.J.T. Allenby: Rings, Fields and Groups; Butterworth-Heinemann
• J.A. Green: Sets and Groups; Chapman and Hall

Online resources:

• Peter Cameron: Notes on a course "Introduction to Algebra" (in particular Chapters 4 & 9, as well as Ch. 5, 6 & 10)