# 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.

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