Math 103A: Modern Algebra I
Winter 2014

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

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.