## Math2108A/Mathematics 3101A

Abstract Algebra I / Algebraic Structures with Computer Applications

**Course Outline****Notice Board**My office hour: Tuesday 10:30-11:30am, Thursday, 3:00-4:00pm, HP4368.

TA: Georgios Tzanakis, office hours: Tuesday 3-4pm, HP 4354

Topics covered in the class.

- Sept. 8: Introduction; Sets, mappings.
- Sept. 13: Mappings, equivalence relation.
- Sept. 15: equivalence classes, division algorithm
- Sept. 20: gcd, Euclidean algorithm, congruence, monoids
- Sept. 22: monoids, deterministic finite automaton.
- Test #1
- Sept. 27: groups
- Sept. 29: more examples of groups, basic properties.
- Oct. 4: Exponents/multiples, subgroups.
- Oct. 6: Cyclic groups.
- Oct. 11: multiplicative subgroups of complex numbers; n-th roots of unity.
- Oct. 13: permutation groups.
- Test #2.
- Oct. 18: permutation groups, alternating groups, Dihedral groups.
- Oct. 20: Dihedral groups, motion group of a cube.
- Oct. 25: motion group of a cube, conjugates, cosets.
- Oct. 27: Lagrange' Theorem and applications.
- Test #3.
- Nov. 1: Lagrange's Theorem, Fermat's Theorem, Euler's Theorem, classical cryptography.
- Nov. 3: RSA, isomorphism.
- Nov. 8: Isomorphism, Cayley's Theorem.
- Nov. 10: Direct products, normal subgroups.
- Test #4
- Nov. 15: Normal subgroups and quotient groups.
- Nov. 17: Quotient groups and fundamental theorem of homomorphism.
- Nov. 22: Rings.
- Nov. 24: Ideals and Integral domain.
- Nov. 29: quotient rings, maximal ideals and prime ideals.
- Dec. 1: Polynomial rings, irreducible polynomials, construction of finite fields.
Tutorials:

- Tutorial #1: Chapter 1: 17, 20, 22, 25.
- Tutorial #2: Chapter 3: #1, 7, 10, 12, 27.
- Tutorial #3: Chapter 3: #32, 40, 43, 47. Chapter 4: #5, 8.
- Tutorial #4: Chapter 4: 32, 37, 43. Chapter 5: 2 (a) (i), 4, 13, 30.
- Tutorial #5: Chapter 6: 5(a), (b), 12. Chapter 7: 7(a), 8(a). Chpater 9: 5.
Suggested Exercises.

- Chapter 1: 1-29.
- Chapter 2: 1, 5, 12-31.
- Chapter 3: 1-32, 33-54.
- Chapter 4: 1-12, 13-43.
- Chapter 5: 1-6, 20-27.
- Chapter 6: 1-6, 11-21.
- Chapter 7: 7-10.
- Chapter 9: 1-11, 14-17, 26-28, 46-48, 52.
**Chapter 10: 1-10.**