Algebraic Structures, Fall 2023
Group ring
© 2021 Laure Bukh
Used with permission
When:
- Mondays, Wednesdays, Fridays 10:00 (Section A)
- Mondays, Wednesdays, Fridays 2:00 (Section B)
Where:
- Baker Hall 235A (Section A)
- Porter Hall A18A (Section B)
What:
The aim of this course is to introduce algebraic structures that pervade mathematics: groups and rings. We will learn what they are, will see many examples, learn how to reason about them. Topics to be covered include permutation groups, abelian groups, cyclic groups, homomorphisms, quotient groups, group actions, group classification, rings, ring homomorphisms, ideals, integral domains, quotient rings, unique factorization domains, principal ideal domains, and fields.
The prerequisites are being comfortable with reading and writing proofs, and a little of bit of linear algebra.
Resources:
- Lecture notes by Samir Siksek. They are well-written, but contain slightly less than what we are going to cover.
- Abstract Algebra by Dummit and Foote. This is an excellent book for self-study, both at the beginner and more advanced levels. A copy is on reserve in the library.
Office hours:
The office hours will be at 2:30pm–3:20pm on Thursdays and at 11am–11:55am on Fridays in Wean 6202. I am also available by appointment.
Discussion forum:
We use Piazza discussion forum.
Course activities:
There will be weekly homeworks, two mid-terms and a final. The mid-terms will take place on September 25th and November 8th. The final exam will be December 11th at 5:30pm in Hamerschlag Hall B131. In case of a final exam conflict, the students are required to inform me and the other involved instructor within one week after the registrar publishes the final exam schedule.
Students are expected to fully participate in the class. Discussions during the lectures are encouraged.
Homework will count for 10% of the grade. The mid-terms will count for 25% each, whereas the final will count for 40%.
Homeworks:
Practice is an integral to learning mathematics. You are encouraged to do as much homework as possible on your own; this way you will learn more. Though collaboration is allowed, you must write the solutions yourself. Turning in solutions that you do not understand will be treated as cheating. In particular, you are allowed to use (with a citation) any source, but only if you have read and understood it.
Homework must be neat. Each word must be readable. Anything that you do not want to be graded must be completely crossed out. If in doubt, either re-write solution from scratch or typeset it in LaTeX. Any solution that fails to be neat will receive 0.
The lowest homework score will not count towards the final grade.
The homework must be submitted via Gradescope.
Exams:
All exams are closed book.
The in-class exams for sections A and B will be different. However, to avoid misplaced expectations and additional stress, I strongly recommend students in section A not to discuss the exam with students in section B.
If for unforeseeable reason you are unable to take one of the exams, contact me as soon as you are able.
Academic integrity:
Violations of academic integrity include, but are not limited to,
- Not writing solutions independently.
- Not disclosing help that you received and resources used.
- Turning in solutions that you do not understand.
- Receiving or providing assistance during an exam.
The default course-level action for violation of academic integrity is an R grade for the course, but more lenient action may be considered depending on the nature of the violation and conduct during the investigation.
If you feel desparate, and are tempted to commit a violation, note that there is probably a better way. Please reach out for support to me, and or to many of the university-wide resources.
Staying sane and healthy:
This is an advanced mathematics course. It is designed to challenge your brain with new and exciting mathematics, not to wear your body down with sleepless nights. Start the assignments early, and get good nutrition and exercise. Pace yourself, for semester is long. If you find yourself falling behind or constantly tired, talk to me.
Lectures:
Section A | Section B | ||
---|---|---|---|
August 28 | Introduction. Groups. Examples. Commutativity. Abelian groups. D_{8}. Symmetry. | Homework #1 | |
August 30 | Notation. Powers. Subgroups. Dihedral groups. GL_{2}(R). Dihedral group as a subgroup of GL_{2}(R). Cyclic groups. | Notation. Powers. Subgroups. Dihedral groups. GL_{2}(R). Classification of subgroups of Z. | |
September 1 | Symmetric group. Cycles. Classification of subgroups of Z. | Symmetric group. Cycles. Dihedral group as a subgroup of GL_{2}(R). Cyclic groups. | |
September 4 | Labor Day | Homework #2 | |
September 6 | Permutations are products of disjoint cycles. Order of a group. Order of an element. Homomorphisms. Examples. Isomorphisms. Examples. Cyclic groups are determined by their order. | Permutations are products of disjoint cycles. Order of a group. Order of an element. Homomorphisms. Examples. | |
September 8 | Properties of homomorphisms and isomorphisms. Heisenberg group. Conjugation. | Isomorphisms. Examples. Cyclic groups are determined by their order. Properties of homomorphisms and isomorphisms. Heisenberg group. Conjugation. | |
September 11 | Subgroup notation. Cosets. Lagrange's theorem. Groups of prime order. Index. Group (Z/nZ)^{*}. Euclidean algorithm. Fermat's little theorem. | Subgroup notation. Cosets. Lagrange's theorem. Groups of prime order. Index. Fire alarm. | Homework #3 |
September 13 | Examples of cosets. Products of groups. Normal subgroups. Kernels. Normality criterion. Quotient groups. SL_{2}(R) | Group (Z/nZ)^{*}. Euclidean algorithm. Fermat's little theorem. Examples of cosets. Products of groups. Normal subgroups. | |
September 15 | Natural projection. Trivial kernels. First isomorphism theorem. Normality is not transitive. | Quotient groups. Kernels. Index-2 subgroups are normal. Normality is not transitive. | |
September 18 | Index-2 subgroups are normal. Groups generated by subsets. | Normality criterion. SL_{2}(R). Trivial kernels. First isomorphism theorem. | Homework #4 |
September 20 | Groups actions. Group actions are homomorphisms. Cayley's theorem. Orbits. Stabilizers. | Natural projection. Groups generated by subsets. | |
September 22 | Examples of groups actions. Restrictions of group actions. Transpositions. Sign of a permutation. | Groups actions. Group actions are homomorphisms. Cayley's theorem. Orbits. | |
September 25 | Test #1 | Homework #5 | |
September 27 | Alternating group. Simple groups. Alternating groups are generated by the 3-cycles. Alternating groups are simple (part I). | Examples of groups actions. Restrictions of group actions. Transpositions. Sign of a permutation. | |
September 29 | Alternating groups are simple (part II). A_{4} is not simple. Converse to Lagrange's theorem fails. A digression on solvable groups and solving polynomial equations. Automorphism group. Inner automorphisms. p-groups. | Alternating group. Simple groups. Alternating groups are generated by the 3-cycles. Alternating groups are simple (part I). | |
October 2 | Orbit-stabilizer theorem. Center. p-groups have subgroups of all conceivable orders. Normalizers. Sylow p-subgroups. Sylow's theorem I (statement). | Alternating groups are simple (part II). A_{4} is not simple. Converse to Lagrange's theorem fails. Automorphism group. Inner automorphisms. Orbit-stabilizer theorem. | Homework #6 |
October 4 | Criterion for the existence of Sylow p-subgroups. Sylow p-subgroups of S_{kp} for small k. Source for the proof | Center. p-groups have subgroups of all conceivable orders. Normalizers. Sylow p-subgroups. Sylow's theorem I (statement). Product of subgroups (part I). | |
October 6 | Fields. Groups GL_{n}(F_{p}). Product of subgroups. | Criterion for the existence of Sylow p-subgroups. Sylow p-subgroups of S_{kp} for small k. Source for the proof | |
October 9 | Product of subgroups II. Normalizers revisited. The other Sylow theorems. | Fields. Groups GL_{n}(F_{p}). Normalizers revisited. Product of subgroups II. | Homework #7 |
October 11 | Applications of Sylow's theorems. Groups of order pq (part I). Semidirect products. | The other Sylow theorems. An application of Sylow's theorems. | |
October 13 | Examples of semidirect products. Groups of order pq (part II). Orthogonal group. Rigid motions of Euclidean space. | Groups of order pq. Semidirect products. | Notes on finite abelian groups |
October 16–20 | Fall break | ||
October 23 | Rings. Examples. Matrices over arbitrary rings. Commutative rings. Center of a ring. Quadratic extensions of Q. Zero divisors. Units. Polynomial rings. Power series. Laurent series. | Examples of semidirect products. Orthogonal group. Rigid motions of Euclidean space. Rings. Examples. Commutative rings. Center of a ring. Quadratic extensions of Q. | Homework #8 |
October 25 | Polynomial rings in several variables. Ring homomorphisms. Evaluation homomorphisms. Polynomial functions. The difference between polynomials functions and polynomials. Kernels. Ideals. Quotient rings. Ideals in Z. Generation of ideals. Sum, product and intersection of ideals. | Polynomial rings. Power series. Laurent series. Polynomial rings in several variables. Ring homomorphisms. Evaluation homomorphisms. Polynomial functions. The difference between polynomials functions and polynomials. Kernels. Ideals. | |
October 27 | Maximal ideals. Complex numbers as a quotient ring. Integral domains. Euclidean domains. Ideals in Euclidean domains are principal. | Quotient rings. Ideals in Z. Generation of ideals. Sum, product and intersection of ideals. Maximal ideals. | |
October 30 | PID. Gaussian integers. Conjugate and norm in quadratic extensions. Gaussian integers are an Euclidean domain. Prime ideals. Irreducible elements. Examples of non-unique factorization. | Complex numbers as a quotient ring. Zero divisors. Units. Integral domains. Euclidean domains. Ideals in Euclidean domains are principal. PID. | Homework #9 |
November 1 | UFD. PID are UFD. Examples. | Gaussian integers. Conjugate and norm in quadratic extensions. Gaussian integers are an Euclidean domain. Prime ideals. Irreducible elements. Examples of non-unique factorization. Generation of rings vs generation of fields. | |
November 3 | GCD. Rings of fractions. Generation of rings vs generation of fields. Content of polynomials. | UFD. PID are UFD. | |
November 6 | Polynomial rings over UFD. | GCD. Rings of fractions. Content of polynomials. | Homework #10 |
November 8 | Test #2 | ||
November 10 | Factorization in Gaussian integers. Sums of two squares. | Polynomial rings over UFD. | |
November 13 | Chinese remainder theorem. Euler's totient function is multiplicative. Lagrange interpolation formula. | Factorization in Gaussian integers. Sums of two squares. | Homework #11 |
November 15 | Fields. Subfields. Extensions. Prime fields. Characteristic. Vector spaces over fields. Degree of an extension. Adding a root. | Chinese remainder theorem. Euler's totient function is multiplicative. Lagrange interpolation formula. | |
November 17 | Algebraic elements. Algebraic extensions. Finite extensions. Minimal polynomials. Degree of iterated extensions I. | Fields. Subfields. Extensions. Prime fields. Characteristic. Vector spaces over fields. Degree of an extension. Adding a root. | |
November 20 | (Non-examinable) Real roots. Sturm's sequences. | Algebraic elements. Algebraic extensions. Finite extensions. Minimal polynomials. | Homework #12 |
November 22–24 | Thanksgiving holiday | ||
November 27 | Degree of iterated extensions. Algebraic numbers. Q-bar. Algebraically closed fields. | Homework #13 | |
November 29 | Irreducibility of low-degree polynomials. Gauss's lemma and irreducibility. Rational root test. Eisenstein's criterion. | ||
December 1 | Straightedge and compass constructions I. | ||
December 4 | Straightedge and compass constructions II. Complex exponentials. Irrationality of e. | ||
December 6 | Separability. Splitting fields (existence only). Finite fields (existence). | ||
December 8 | Finite fields (uniqueness). Irreducible polynomials over F_{p}. |