Math 370 Syllabus Outline

Fall 2009

Week 1: Mathematical induction, review of integers, definition of groups, rings, devision rings, fields. Examples: general linear groups and special linear groups, Hamiltonian quaternions.
Weeks 2: Equivalence relations, modular arithmetic, subgroups, subgroups generated by a subset, cyclic groups, properties of Euler's &phi function, symmetric groups in n letters, Definition of homomorphisms between groups.
Week 3: Left and right cosets, normal subgroups, quotient groups, kernel and image of a homomorphism between groups, correspondence between subgroups of the source and target of a homomorphism. Examples, homomorphisms from a cyclic group to a group, adjoint homomorphism of a group.
Week 4: More examples of groups (quaternion groups, dihedral groups, finite Heisenberg groups, generalized quaternion groups, orthogonal groups), sign of a permutation, group ring of a finite group, product of groups, definition of ideals, vector spaces, vector subspaces, quotient vector spaces.
Week 5: Linear independence, basis and dimension of a vector space. linear transformations, matrix representation of linear transformations.
Week 6: Change of bases, normalizer and centralizer of a subgroup, decomposition of a finite group into conjugacy classes, the class equation.
Week 7: finite p-groups, group actions (groups as abstract symmetries), stabilizer subgroups, orbits, decomposition into orbits, adjoint action as example.
Week 8: midterm exam (in class), more examples of group actions, translation action on G/H.
Week 9: Sylow's theorems, examples, classification of finite groups of small order.
Week 10: Linear operators, eigenvalues, eigenspaces, dual vector spaces, transpose of a linear transformation, the Euclidean algorithm on a polynomial ring over a field.
Week 11: Principal ideal domains, unique factorization, gcd in a polynomial ring.
Week 12: Primary decomposition for a linear operator on a finite dimensional vector space.
Week 13: Reformulation in terms of matrices, canonical forms.