Suggested Syllabus for Math 206

Groups, rings and modules are the basics of understanding the algebra of operations. Elementary linear algebra is the basic tool for understanding one operation acting on a vector space is. Understanding how two (or more) simultaneous linear operations intereract on a vector space has many applications. The material for this integrates the use of algebra and geometry. The first two quarters will integrate the mathematical story that comes from considering the groups from orthogonal and hyperbolic geometry that justify the basics of groups and modules. The course presentation will revolve around illustrating examples rather than abstraction. A major course goal will be to develop students' problem solving skills.

The first two quarter special topics include the following:

Quarter 1:

1. Geometry behind row reduction of a matrix and space of solutions of linear equations.

2. Uniqueness of Row Reduction as an example of representatives of an equivalence class.

3. Change of basis in a vector space.

4. Interpreting the group action of conjugation by invertible matrices

5. Ring of endomorphisms and action of a group on a set

6. Normal subgroups and semi-direct product

7. Approaching Group Theory through Linear Algebra

8. Invariant Subspaces, Characteristic polynomials and eigenvectors.

9. Semi-direct Product for translations and rotations

10. Symmetric and alternating groups and the determinant function.

11. Cramer's rule in any commutative ring.

Quarter 2:

1. Modules over a principal ideal domain.

2. Interpretation of operations from upper triangular and diagonal matrices.

3. Presentation of matrices from special subgroups of the general linear group.

4. Fermat's little Theorem and the group of affine transformations of the integers modulo n.

5. Classification of finitely generated modules over a principal ideal domain.

6. Application 1: Classification of abelian groups and lattices in Rⁿ.

7. Application 2: Similarity classes of matrices: Jordan and rational canonical form.

8. Subgroups of the general linear group acting on a vector space over a finite field.

9. Group actions on a set and the classification of permutation representations.

10. The Sylow theorems and p-subgroups of the symmetric groups.

11. Dual spaces, the orthogonal group and Gram-Schmidt diagonalization,

12. Unitary operators and their action on Hermitian operators.

Special topics for the 3rd quarter: The material of the first two quarters allows many applications though interpreting algebraic operations on a set through group actions using linear algebra. Several applications following from the previous material include the following.

1. Classification of matrices commuting with a given matrix.

2. Stability of matrix solutions.

3. Fundamental Theorem of algebra by applying the 1st Sylow theorem.

4. Theory of the Galois group attached to a polynomial in one variable.

5. Interpretation of the solvability of a polynomial equation.

5. Elementary representation theory of finite groups.

6. An introduction to simple groups and their applications.

7. Elementary multilinear algebra.

8. Unique factorization domains.

9. Affine algebraic spaces and affine groups.

10. Ideal class groups of a ring.

11. An introduction to elementary algebraic number theory.