MAT 312/AMS 351
Applied Algebra
Summer 2012

July 10, 12, 17, 19, 24, 26, 31
August 2, 7, 9, 14, 16

Elementary number theory (4 lectures)
-greatest common divisor, division theorem (Euclidean algorithm), prime factorization
-sets, relations, equivalence classes, inclusion-exclusion principle
-congruence classes, linear congruences, Chinese remainder theorem
-group and ring properties of congruence classes, Fermat's theorem, Euler's theorem
(Humphreys and Prest 1.1, 1.3, 1.4, 1.5, 1.6, 2.1, 2.2, 2.3)

Groups and Symmetry (4 lectures)
-brief review of Euclidean geometry, classification of rigid motions
-planar transformation groups, cyclic and dihedral groups, the abstract definition of group
-conjugation, isomorphism, Lagrange's theorem
-homomorphisms, quotients, actions, orbits, Burnside's theorem
-the symmetric group, matrix groups
(Duzhin and Chebotarevsky, Ch. 2, 3, 4, 5, Humphreys and Prest 4.1, 4.2, 4.3)

Applications (4 lectures)
-public key cryptography; RSA
-binary error-correcting and error-detecting codes
-molecular point symmetry, group characters
(Humphreys and Prest 1.6, 5.4)