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)