School of Mathematics, Trinity College

Course 1214 - Introduction to group theory 2013 (click for more information)

JF Mathematics, JS Theoretical Physics, JF Two-subject Moderatorship

Lecturer Dmitri Zaitsev

Annual Examinations: The format will be the same as in the year 2011-2012 exams, which can be considered sample papers. Credit will be given for the best 3 questions out of total 4 questions. The theoretical questions will be within the scope of the current course and the practical problems within the scope of the current homework. The final mark is 90% exam and 10% homework.

Lecture Notes in PDF (to be continued) are only meant to supplement the material and older lecture notes

Problem Sheets in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5 Sheet 6 Sheet 7 Sheet 8 Sheet 9

Problem Solutions in PDF: Sheet 1 Sheet 2 Sheet 3 Sheet 4 Sheet 5 Sheet 6 Sheet 7 Sheet 8 Sheet 9

Course outline (to be continued):

Sets, their unions, intersections, differences, direct (or cartesian) products. Subsets. Maps between sets, injective, surjective and bijective maps. Images and preimages of subsets. Composition of maps. Identity map and Inverse of map.

Binary operations on sets. Associativity, multiplicativity. Identity and inverse elements with respect to a binary operation. Groups, semigroups, monoids. Cayley table of a group. Direct products of groups. Subgroups. Intersections of subroups. Generators of a subgroup.

Permutation group of a set (the group of all bijective self-maps). Symmetric group Sn. Parity (sign) of a permutation, even and odd permutations. Alternating subgroup An of Sn. Group of Isometries. Matrix groups GLn, SLn, On, SOn, Un, SUn.

Integer division with remainder. Additive subgroups of Z. Greatest common divisor. Euclidean algorithm. Unique prime factorization. Binary relations, equivalence relations, partitions. Congruence relation and classes of integers modulo n. The set of congruence classes Zn modulo n as additive group and multiplicative monoid. Multiplicative group Zn*.

Cosets of a subgroup in a group. Lagrange's Theorem.

Group homomorphisms and isomorphisms. Kernel of homomorphism. Normal subgroup. Quotient group modulo normal subgroup. First isomorphism theorem.

Chinese remainder theorem. Group actions on a set. Stabilizer. Orbits.

John R. Durbin, Modern Algebra: An Introduction, John Wiley & Sons.
B.L. van der Waerden, Algebra, Volume 1. (AMS Notices Introduction)
Serge Lang, Undergraduate Algebra. Springer.

Some links.

Old courses homepages:
Introduction to group theory MA1214 2012 by Rudolf Tange with exercise sheets and solutions.
Introduction to group theory MA1214 2011 by Colm Ó Dúnlaing with Web Notes, Quiz answers and Extra problems.
Course 111 - Algebra 1996-97 by David Wilkins with Lecture Notes.

For exam-related problems look in TCD past examination papers and Mathematics department examination papers.

Student Counselling Service

I will appreciate any (also critical) suggestions that you may have for the course. Let me know your opinion, what can/should be improved, avoided etc. and I will do my best to follow them. Feel free to come and see me if and when you have a question about anything in this course. Or use the feedback form from where you can also send me anonymous messages.