Fall 2015

**Instructor:** Brendon Rhoades

**Instructor's Email:** bprhoades (at) math.ucsd.edu

**Instructor's Office:** 7250 APM

**Instructor's Office Hours:** 10:00-11:00am MWF or by appointment

**Lecture Time:** 9:00-9:50am MWF

**Lecture Room:** 7421 APM

**Final Exam Time:** TBA

**Final Exam Room:** TBA

** Grading:**

Homework: 30%

Midterm (10/23, in class): 30%

Final Exam: 40%

** Textbook:** *Abstract Algebra, 3rd Edition* by
David Dummit and Richard Foote

** Lectures:**

**9/25:** Section 1.1. Group axioms, groups of numbers,
symmetric group, integers modulo n. Abelian groups, orders of group elements.
Section 1.2. Dihedral groups. Generation. Presentations.
Two presentations of the dihedral group.
Section 1.3. Cycle notation for permutations. Multiplication
convention.
Section 1.4. Fields. Matrix groups (GL, SL).
Section 1.5. Quaternions Q_8.

**9/28:** Section 1.6. Homomorphisms/Isomorphisms.
Examples. Section 1.7. Group actions on sets. Examples:
dihedral groups, matrix groups, permutation groups.
Kernels, orbits.
Section 2.1. Subgroups. Examples. Section 2.2.
Stabilizers, centralizers, normalizers, centers.
Section 2.3. Cyclic subgroups, cyclic groups. Subgroups of
cyclic groups. Section 2.4. Subgroups generated by subsets.

**9/30:** Group homomorphisms from group actions.
Section 3.1. Normal subgroups. Examples and non-examples.
Cosets and G/H. The natural "multiplication" formula on G/H is
well-defined iff H is normal in G, in which case G/H is a group.
Section 3.2. Lagrange's Theorem and some corollaries. Formula
for |HK|. Section 3.3. First isomorphism theorem. A homomorphism
f is injective iff ker(f) = 1. Second isomorphism theorem.

**10/2:** Section 3.3. Third isomorphism theorem. Fourth isomorphism
theorem. Criterion for homomorphisms out of G/N to be well defined.
Section 3.4. Simple groups. Composition series.
Examples: Z_n, S_3, D_{2n}.
Jordan-Holder theorem. Solvable groups. Examples.
Conjugacy within S_n via cycle notation.

**10/5:** Section 3.5. The sign homomorphism and the alternating
group. Transpositions and permutation parity.
Section 4.1. Permutation representation. Faithful actions.
Examples. Transitive actions. Orbit/Stabilizer Theorem.
Examples.

**10/7:** Section 4.2. Left multiplication action of G on G/H.
Cayley's Theorem. If G is finite and [G:H] = smallest prime dividing |G|,
then H is normal. Conjugation actions. Class equation.
p-groups. Every p-group has a nontrivial center. Classification
of groups of order p^2. The alternating group A_5 is simple.

**10/9:** Section 4.4. Automorphisms. Inner automorphisms.
Characteristic subgroups. Aut(Z_n), Aut(Z_p x ... x Z_p).
Section 4.5. Cauchy's Theorem. Introduction to Sylow theorems.

**10/12:** Section 4.5. Proof of the Sylow theorems.
Application: There is no simple group of order 1365.

**10/14:** Section 4.5. Applications of the Sylow theorems.
Groups of order pq and (p^2)q. Section 4.6. A_n is simple for n > 4.
Section 5.1. Review of direct products of groups.
Section 5.2. The Fundamental Theorem of Finitely Generated Abelian
Groups (statement only!).

**10/16:** Section 5.4. [x,y], [A,B], G' = [G,G].
Theorem on recognition of direct products G = H x K. Section 5.5.
Intro to semidirect products.

**10/19:** Section 5.5. More on semidirect products. The external
semidirect product of H and K. Application to groups of order 39.
Dihedral groups, signed permutation groups, and affine linear groups
as semidirect products.

**10/21:** Semidirect product wrap-up. Section 6.3.
The free group F(S) on a set S and its universal property.
Statements (no proofs!): every subgroup of a free group is free,
the free group on {x, y} contains a free group of any finite rank
as a subgroup.
Presentations G = < S | R >. Word problems. A presentation
of Q_8.

**10/23:** Midterm.

**10/26:** Section 6.1. Properties of p-groups. The upper
central series of a group G. Nilpotent groups. Examples with
dihedral groups.

**10/28:** Section 6.1. Structure theorem for finite nilpotent
groups. The lower central series of a group G. A group is nilpotent
iff its lower central series reaches 1.
The derived series of a group G. A group is solvable
iff its derived series reaches 1.

**10/30:** Section 7.1. Ring axioms. Examples.
Units, zero divisors, unit group. Fields, division rings, integral
domains. Hamiltonians. Section 7.2. R[x] for a ring R.

**11/2:** Section 7.2. The ring M_n(R) of matrices over a ring R.
The group ring RG for a ring R and a finite group G.
Section 7.3. Ring homomorphisms. Kernels.
Ideals and quotient rings.
First Isomorphism Theorem for rings. Examples.
The ideal (A) generated by a subset A of a ring. Finitely generated ideals.
Principal ideals.

**11/4:** Section 7.3.
The ideal (2,x) in Z[x] is not principal. Section 7.4.
Maximal ideals.
For R commutative, I is maximal iff R/I is a field.
Zorn's Lemma. For any proper ideal I, there exists maximal
ideal M containing I.
Prime ideals.
For R commutative, and ideal I is prime iff R/I is an integral domain.
Examples.

**11/6:** Section 7.5. The fraction ring D^{-1}R (where R is a
commutative ring with 1, D is a closed-under-multiplication subset
of R containing 1 and not containing 0 or zero-divisors). Examples.
The universal property of D^{-1}R. The field of fractions of an integral
domain R. Section 7.6. Ideal sum I+J and product IJ. Comaximal ideals.
The Chinese Remainder Theorem.

**11/9:** Section 8.1. Norms on domains. Euclidean domains.
Examples: Z, F[x], Z[i], fields.
Non-examples: Z[x], F[x,y].
In a Euclidean domain, every ideal is principal.
Greatest common divisors. The Euclidean algorithm.
Section 8.2. Principal ideal domains. In a PID,
every nonzero prime ideal is maximal.

**11/13:** Section 8.2. GCDs in PIDs. Section 8.3.
Irreducibles and primes in integral domains. Examples. Definition of
a UFD. Examples and non-examples. In a UFD, primes and irreducibles coincide.

**11/16:** Section 8.3. Lemma: In a PID, primes and irreducibles
coincide. Theorem: Every PID is a UFD. Definition of a Noetherian ring.
In a Noetherian domain, any nonzero nonunit factors into a finite
product of irreducibles (but not necessarily uniquely -- Z[sqrt(-5)] is
a counterexample). Z[2i] is not a UFD. Section 9.1. Polynomial rings R[x].
If I is an ideal in R, then (I)_{R[x]} = I[x] and
R[x]/(I) is isomorphic to (R/I)[x].

**11/18:** Section 9.2. Polynomial long division.
Section 9.3. Let R be a UFD and let F be its fraction field.
Gauss' Lemma: If p(x) = A(x) B(x) for nonconstant A(x), B(x) in F[x],
there exist u, v in F such that p(x) = a(x) b(x), where
a(x) = u A(x) and b(x) = v B(x) lie in R[x]. If R is a UFD and p(x)
is in R[x] with g.c.d. of its coefficients = 1, then p(x) is reducible in
R[x] iff p(x) is reducible in F[x].
Theorem: If R is a UFD, so is R[x].

**11/20:** Section 9.4. Irreducibility criteria for polynomials
in R[x], for R a domain. If f(x) is irreducible in (R/I)[x] for some
proper ideal I and f(x) is monic and nonconstant, then f(x) is irreducible
in R[x]. Eisenstein's Criterion. Rational roots test.
Irreducibility in low degree.

**11/23:** Section 9.5. Structure of rings F[x]/(f(x)).
A polynomial f(x) in F[x] has at most deg(f) roots.
Application: If G is any finite subgroup of the unit group
of a field, then G is cyclic.
Section 9.6. Noetherian rings. Examples and non-examples.

**11/30:** Section 9.6. Proof of Hilbert's Basis Theorem.

** Homework Assignments: **

Homework 1, due 10/5/2015.

Homework 2, due 10/12/2015.

Homework 3, due 10/19/2015.

Homework 4, due 10/30/2015.

Homework 5, due 11/6/2015.

Homework 6, due 11/18/2015.

Homework 7, due 12/2/2015.