Winter 2014

**Instructor:**
Brendon Rhoades

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

**Instructor's Office:** 7250 APM

**Instructor's Office Hours:** MWF 4:00-5:00 pm

**TA:** Quang Bach

**TA's Email:** qtbach (at) ucsd.edu

**TA's Office:** 5720 APM

**TA's Office Hours:** F 2:00-3:00 pm

**Lecture Time:** MWF 11:00-11:50 am

**Lecture Room:** 147 SEQUO

**Discussion Time:** Tu 8:00-8:50 am

**Discussion Room:** B412 APM

**Final Exam Time:** 11:30 am - 2:29 pm, 3/17/2014

**Syllabus:** Please read carefully

**Lectures:**

**1/6:**
Syllabus and administrivia.
Chapter 0. The integers Z = {...,-2,-1,0,1,2,...}.
Divisors, prime numbers, and the Division Theorem. Greatest common divisors.
gcd(a,b) is the minimum positive integer linear in the set
{as+bt : s, t in Z}.

**1/8:**
Chapter 0. Euclid's Lemma and unique prime factorization.
Least common multiples. gcd(a,b) and lcm(a,b) in terms of prime
factorizations. Equivalence relations and equivalence classes.

**1/10:**
Chapter 1. Symmetries of a square: the dihedral group D_4. The
Cayley table of D_4. Properties of multiplication in
D_4 (closure, identity, inverses, associativity).
Symmetries
of the regular n-gon (the dihedral group D_n).

**1/13:**
Chapter 2. Binary operations. Examples and non-examples.
The definition of a group. Examples:
(symmetry groups such as D_n,)
the (integers, rational numbers,
real numbers) under addition, the set {1,-1,i,-i} under multiplication,
the positive rational numbers under multiplication.
Non-examples: The integers under multiplication.

**1/15:**
Chapter 2. Identities and inverses in groups are unique. More examples
and non-examples of groups: mxn real (or rational, or complex, ...)
matrices under matrix addition, R^n, the set Z_n = {0,1,...,n-1}
under addition mod n, the general linear group GL(n,R) of
nxn real invertible matrices under matrix multiplication (or GL(n,Q), etc.).

**1/17:**
Chapter 2. Left and right cancellation in groups. (ab)^{-1} = b^{-1} a^{-1}.
More examples and non-examples of groups: the group
U(n) under multiplication mod n, the group of complex numbers under addition,
the group of nonzero complex numbers under multiplication,
the group of complex n^{th} roots of unity under multiplicaiton,
the group of translations of R^2 under functional composition,
the special linear group SL(n,F) (where F = R, C, or Q).

**1/22:**
Chapter 3. The order of a group and the order of an element of a group.
Subgroups.

**1/24:**
Chapter 3. The one- and two-step subgroup tests.

**1/27:** Chapter 3. If G is a group and H is a nonempty finite
subset of G which is closed under multiplication, then H is a subgroup.
The cyclic group < g > generated by an element g in G. The subgroup < S >
generated by a subset S of a group G.

**1/29:** Chapter 3. The center Z(G) of a group. The centralizer
C(a) of an element a in a group. Chapter 4. The definition of a cyclic
group. Examples: Z, Z_n, U(10). Non-examples: U(8), any non-Abelian group.
Criterion for when a^i = a^j in a group. |a| = |< a >|.

**1/31:** Midterm 1.

**2/3:** Chapter 4. If |a| = n, then < a^k > = < a^gcd(k,n) >.
Criterion for < a^i > = < a^j >. Subgroup lattices. Statement of the
characterization of the subgroup lattices of cyclic groups. Anonymous
course feedback.

**2/5:** Chapter 4. Proof of the Fundamental Theorem of Cyclic Groups.
Chapter 5. Permutations and the symmetric group S_n. Permutation notation:
functional, two-line, one-line, and cycle. Multiplying permutations
in cycle notation. |S_n| = n!.

**2/7:** Chapter 5. The identity and inverses in cycle notation.
The order of a permutation from cycle notation. Writing permutations
as products of 2-cycles. Even and odd permutations. The alternating
group A_n.

**2/10:** Chapter 6. The definition of an isomorphism phi: G to H.
The reals under addition are isomorphic to the positive reals under
multiplication. Every cyclic group is isomorphic to either Z or Z_n.
The map f: R to R, f(x) = x^5 is not an isomorphism. U(10) is not isomorphic
to U(12). Basic properties of isomorphisms.

**2/12:** Chapter 6. More basic properties of isomorphisms. Cayley's
Theorem: Every group is isomorphic to a group of permutations.
The group Aut(G) of automorphisms of a group.
Conjugation; the subgroup Inn(G) of inner
automorphisms of a group.

**2/14:** Chapter 6. Aut(Z_n) is isomorphic to U(n). Chapter 7.
Coset notation: gH, Hg, gHg^{-1}. Some examples. Basic properties
of cosets. Lagrange's Theorem.

**2/19:** Chapter 7. The index [G:H] of a subgroup H of G;
[G:H] = |G|/|H|. g^{|G|} = e. Every group of prime order is cyclic.
Fermat's Little Theorem.
Formula for |HK|.
The orbit orb_G(i) and the stabilizer stab_G(i).
The Orbit-Stabilizer Theorem.

**2/21:** Chapter 8. The external direct product G_1 + ... + G_n of
groups G_1, ..., G_n. Formula for element orders in direct products.
Criterion for direct products to be cyclic. Z_{st} = Z_s + Z_t iff
gcd(s,t) = 1. If gcd(s,t) = 1, then U(st) = U(s) + U(t).

**2/24:** Chapter 9. Normal subgroups, criterion for normality.
{e, (1,2)} is not a normal subgroup of S_3. A_n is a normal subgroup of
S_n. Any subgroup consisting only of rotations is a normal subgroup of D_n.
Z(G) is a normal subgroup of G. Any subgroup of an Abelian group is normal.
If H, K are subgroups of G with H normal, then HK is a subgroup of G.
The set of left cosets G/H = {gH : g in G}.
THEOREM: G/H is a group under (gH)(g'H) = (gg')H if and only if
H is normal in G.

**2/26:** Chapter 9. Examples of quotient groups.
Z/nZ, S_n/A_n, Z_n/< k >. G is Abelian iff G/Z(G) is cyclic.
G/Z(G) is isomorphic to Inn(G).

**2/28:** Midterm 2.

**3/3:** Chapter 9. The internal direct product G = H x K. If
G = H x K, the G is isomorphic to the external direct product
H + K. D_6 is isomorphic to S_3 + Z_2. The internal direct product
of n subgroups G = H_1 x ... x H_n. Classification of groups of
order p^2, where p is prime.

**3/5:** Chapter 10. Group homomorphisms. Kernels.
Basic properties of homomorphisms. The First Isomorphism Theorem:
If f: G ---> H is a homomorphism of groups, then Ker(f) is a normal
subgroup of G and G/Ker(f) is isomorphic to the image f(G).

**3/7:** Chapter 10. Examples of the First Isomorphism
Theorem. Z/< n > is isomorphic to Z_n. GL(n,C)/SL(n,C) is isomorphic to
the multiplicitive group of non-zero complex numbers. If N is a normal subgroup
of G, the canonical projection G ---> G/N is a surjective homomorphism
with kernel N. The evaluation homomorphism R[x] ---> R given by sending
f to f(3). Centralizers, normalizers, and the ``N/C Theorem".

**3/10:** Chapter 11. The Fundamental Theorem of Finite Abelian
Groups. If G is a finite Abelian group and m | |G|, then
there exists a subgroup of G of order m.

**3/12:** Review.

**3/14:** Review.

** Homework:**

Homework 1, due 1/10/2014.

Homework 2, due 1/17/2014.

Homework 3, due 1/24/2014.

Homework 4, due 2/7/2014.

Homework 5, due 2/14/2014.

Homework 6, due 2/24/2014.

Homework 7, due 3/7/2014.

Homework 8, due 3/14/2014.

**Midterms:**

Practice Midterm 1 and
Solutions.

Midterm 1 and
Solutions.

Midterm 1 Score Distribution: 100, 100, 97, 89, 87,
86, 85, 85, 83, 82, 81, 74, 74, 73, 72, 72, 71, 67,
67, 67, 65, 64, 63, 63, 62, 61, 59, 59, 58, 56, 55,
53, 51, 51, 46, 42, 42, 41, 40, 39, 27, 26, 21.

Mean: 64, Median: 64, Standard Deviation: 19.32

Practice Midterm 2 and
Solutions.

Midterm 2 and
Solutions.

Midterm 2 Score Distribution: 90, 88, 81, 76, 73, 71, 70, 68, 68,
66, 65, 64, 63, 62, 62, 62, 60, 58, 56, 54, 54, 53, 52, 51, 50,
47, 47, 44, 43, 41, 40, 39, 38, 37, 37, 36, 35, 21.

Mean: 56, Median: 55, Standard Deviation: 15.45

Practice Final Exam and
Solutions.