MATH 4441/6441– Content Standards
Title: Abstract Algebra I
Prerequisite: Math 3435 with grade C or higher.
Matrix representations of linear transformations, similarity; inner product spaces and orthogonality, least squares problems; eigenvalues, diagonalization, Hermitian matrices, quadratic forms, positive definite matrices, and nonnegative matrices.
Goals
To provide students with a solid background in abstract algebra including an axiomatic approach to algebraic structures, groups, permutations, homomorphisms, and factor groups.
Content Standards
The following standards are offered as guidelines for assessing student progress, judging the effectiveness of instructional programs, and developing curricular units. The subject matter outlined in these standards represents the minimum knowledge in which a student should demonstrate proficiency at the successful completion of the course.
1. Elements of set theory.
2. Equivalence and order relations.
3. Semigroups, monoids.
4. Groups, subgroups, normal subgroups, Lagrange's theorem. Construction of the integers.
5. Homomorphisms, isomorphisms, kernel, image.
6. Quotient groups.
7. Isomorphisms theorems for groups.
8. Center, normalizer, centralizer.
9. Direct products.
10. Cauchy's theorem.
11. Permutation groups.
12. Structure of finitely generated abelian groups (time permitting).
13. Action of groups on sets, Sylow theorems, semidirect products (time permitting).
.A higher level of proficiency will be required for those enrolled in Math 6441.