Schedule of lectures

We plan to cover most of Chapters 1-9 of the text. The following schedule will be updated as we go with information about what we covered on particular days.

• 9/23/16 Chapters 1-3. Review of basic group theory I: definition of a group. Basic examples of groups: the symmetric group of a set, groups of numbers under +, multiplicative groups of numbers, the matrix group GL_n(F) for a field F, quaternion group, dihedral groups, integers mod n. Subgroups.

• 9/26/16 Chapters 1-3. Review of basic group theory II: homomorphisms and isomorphisms. Cosets and Lagrange's theorem. Normal subgroups and factor groups. The 1st isomorphism theorem.
• 9/28/16 Chapters 1-3. Review of basic group theory III: The 2nd-4th isomorphism theorems. Subgroup generated by a subset. Cyclic subgroups and cyclic groups. All cyclic groups are isomorphic to Z or Z/nZ.
• 9/30/16 Section 6.3. The free group on a set X. The universal property of a free group. Presentations. Example: the dihedral group D_{2n} is presented by two generators a, b with relations a^n = e, b^2 = e, ba = a^{-1} b.

• 10/3/16 More on presentations. The presentation of the dihedral group helps to compute endomorphisms of D_{2n}. It is hard to tell if a presented group is trivial or not. Section 4.1. Group actions. Basic examples of actions.
• 10/5/16 More on group actions. Orbits and the orbit stabilizer theorem. The action on left cosets. Application: a subgroup H of index equal to the smallest prime dividing |G| is normal. Burnside's counting lemma.
• 10/7/16 Example of burnside counting. Review of the symmetric group S_n. Every element of S_n is a product of disjoint cycles in an essentially unique way. Conjugacy classes in S_n correspond to distinct cycle shapes. The number of conjugacy classes is given by the partition function.

• 10/10/16 The alternating group A_n. Proof that A_n is simple for n at least 5.
• 10/12/16 Automomorphisms. Inner and outer automorphisms. Inn(G) is isomorphic to G/Z(G). Examples of calculating automorphism groups using presentations. Automorphisms of Z_n. Automorphisms of Z_p times Z_p as matrix groups.
• 10/14/16 Statement and proof of the Sylow theorems.

• 10/17/16 Applications of the Sylow theorems. Groups of order pq and pq^2. Using counting methods to show a group must have at least one normal Sylow subgroup. Characteristic subgroups. If H is characteristic in K and K is normal in G, then H is normal in G. Groups of order 105 have normal Sylow 5 and Sylow 7.
• 10/19/16 Recognizing internal direct products. A group with all of its Sylow subgroups normal is an internal direct product of its Sylow subgroups. Fundamental theorem of Abelian groups (no proofs).
• 10/21/16 Semidirect products. Recognizing semidirect products. Examples: D_{2n} is a semidirect product of Z_n and Z_2; all groups of order pq are semidirect products.

• 10/24/16 More examples of semidirect products. Using semidirect products to classify groups of low order.
• 10/26/16 Commutators and the commutator subgroup. Solvable groups. S_n is solvable only for n less than or equal to 4. The derived series. Subgroups and factor groups of solvable groups are solvable. If N is normal in G and G/N and N are solvable, then G is solvable.
• 10/28/16 in-class midterm

• 10/31/16 Nilpotent groups. Examples. Central series. The upper central series. Finite p-groups are nilpotent. Normalizers grow in Nilpotent groups. A finite nilpotent group is the same as a finite group with all of its Sylow subgroups normal, i.e. such that the group is the direct product of its Sylow subgroups. The lower central series.
• 11/2/16 Frattini's argument. A finite group is nilpotent if and only if every maximal subgroup is normal. Big theorems about solvable groups we won't prove. Introduction to rings: Definitions, basic examples, matrix rings, the zero ring.
• 11/4/16 More on rings: polynomial rings, group rings, direct products of rings. Degree in a polynomial ring. If R is a domain, so is R[x]. The group ring CG of a finite group G over the complex numbers C is isomorphic to a product of matrix rings over C (no proof). Ideals, factor rings. First isomorphism theorem for rings.

• 11/7/16 3rd and 4th isomorphism theorems. Examples of ideals. Zorn's lemma. Using Zorn's lemma to prove that every ring has a maximal ideal.
• 11/9/16 Localization. Multiplicative systems X in a ring R. The definition of the localization RX^{-1}. Universal property of the localization. Examples.
• 11/11/16 Veteran's Day Holiday (no class)

• 11/14/16 Prime and Maximal ideals. An ideal P is prime if and only if R/P is a domain, and an ideal M is maximal if and only if R/M is a field. Maximal ideals are prime. Examples of prime and maximal ideals. Two special cases of localization: localizing at the powers of an element f, and localizing at the complement of a prime ideal.
• 11/16/16 Chinese remainder theorem. Euclidean domains. Z and F[x], F a field, are Euclidean domains.
• 11/18/16 Quadratic integer rings. Z[i] is Euclidean. PIDs. A Euclidean domain is a PID, and moreover a nonzero ideal is generated by any element of minimal norm among nonzero elements in the ideal. Gcds.

• 11/21/16 Associates, irreducible, and prime elements. UFDs. prime elements are irreducible and the converse holds in a PID. Noetherian rings. PIDs are noetherian. Characterizations of noetherian rings.
• 11/23/16 Proof that a PID is a UFD, in fact, every noetherian domain such that irreducible elements are prime is a UFD. In a UFD, gcd's exist. In a UFD, irreducible elements are prime. Examples. Multivariate polynomial rings. R[x] is not a PID if R is a PID which is not a field, but R[x] is a UFD for any UFD R (next time)
• 11/25/16 Thanksgiving Holiday (no class)

• 11/28/16
• 11/30/16
• 12/2/16

• Tu 12/6/16 Final Exam 11:30am-2:30pm