## Math 120A, Fall 2014

• Title: Introduction to Abstract Algebra (Group Theory)
• Instructor: Trevor Wilson (twilson@math.uci.edu)
• Lectures: 11:00–11:50 a.m. MWF in MSTB 122
• Office hours: 1:30–3:30 p.m. on Mondays, or by appointment, in 510V Rowland Hall
• Text: A First Course in Abstract Algebra, 7th edition, by John B. Fraleigh

### Course description

We will study the group axioms, permutation groups, isomorphisms and homomorphisms of groups, quotient groups, and more advanced topics if we have time. Because the subject is rather abstract, definitions and proofs will play a central role in the course.

### Homework

Homework exercises will be assigned after each lecture and posted in the schedule below. The homework assigned in a given week will usually be due on the following Tuesday in your discussion section. Late homework will not be accepted. Your lowest homework score will not be used in calculating your grade in the class.

You may collaborate on the homework with other students and seek help from any source. As always, you must credit your collaborators and cite your sources.

### Quizzes

Quizzes will usually be given on Tuesday and cover the same topics as the homework due on that day. Make-up quizzes will not be given. Your lowest quiz score will not be used in calculating your grade in the class.

### Midterm exam

Sample midterm exam and solutions

### Final exam

Sample final exam and solutions

### Schedule (subject to change)

Week 1:
• Oct. 3 (F) Section 0 exercises 3, 11, 12adf; Section 1 (no exercises for section 1)
• Oct. 6 (M) Section 2 exercises 12, 16, 23ab
• Oct. 8 (W) Section 2 exercises 5, 8, 34
• Oct. 10 (F) Section 3 exercises 4, 5, 6
Week 2:
• Oct. 13 (M) Section 3 exercises 30, 32, 33ab
• Oct. 15 (W) Section 3 exercises 17ab, 22; Section 4 exercise 3. Hint for 3.17ab: write $a \ast b$ as $\phi(\phi^{-1}(a \ast b))$ or $\phi^{-1}(\phi(a \ast b))$ and then try to express this in terms of multiplication "$\cdot$" and the functions $\phi$ and $\phi^{-1}$, which you know.
• Oct. 17 (F) Section 4 exercises 8, 14, 20. In exercise 14 we leave $n$ fixed when forming the structure. So it asks: do $1 \times 1$ diagonal matrices with the stated property form a group under multiplcation? Do $2 \times 2$ diagonal matrices with the stated property form a group under multiplcation? And so on. It turns out that the same argument works for all $n$, so we state and prove it for general $n$. For exercise 20b, recall that $U_4$ denotes the group of complex fourth roots of unity: $\{1,i,-1,-i\}$ under complex multiplication. In exercise 20c, "structurally the same as" means "isomorphic to". Proving associativity by checking all triples of elements is tedious, so we can use the shortcut of finding an isomorphism with a structure whose operation is already known to be associative.
Week 3 (homework set 3 and solutions):
• Oct. 20 (M) Sections 4, 5
• Oct. 22 (W) Section 5
• Oct. 24 (F) Section 6 (Cyclic Groups)
Week 4 (homework set 4 and solutions):
• Oct. 27 (M) Section 6
• Oct. 29 (W) Section 8 (Groups of Permutations)
• Oct. 31 (F) Section 8
Week 5 (no homework was assigned this week):
• Nov. 3 (M) Review
• Nov. 5 (W) Midterm Exam
• Nov. 7 (F) Section 9 (Orbits, Cycles, and the Alternating Groups)
Week 6 (homework set 5 and solutions):
• Nov. 10 (M) Section 9
• Nov. 12 (W) Section 9
• Nov. 14 (F) Section 9
Week 7 (homework set 6 and solutions):
• Nov. 17 (M) Section 10 (Cosets and the Theorem of Lagrange)
• Nov. 19 (W) Section 10
• Nov. 21 (F) Section 11 (Direct Products and Finitely Generated Abelian Groups)
Week 8 (homework set 7 and solutions):
• Nov. 24 (M) Section 11
• Nov. 26 (W) Section 13 (Homomorphisms)
• Nov. 28 (F) Thanksgiving Holiday
Week 9 (homework set 8 and solutions):
• Dec. 1 (M) Section 13
• Dec. 3 (W) Section 14 (Factor Groups)
• Dec. 5 (F) Section 14
Week 10 (practice problems and solutions):
• Dec. 8 (M) Section 15 (Factor Group Computations and Simple Groups)
• Dec. 10 (W) Section 15
• Dec. 12 (F) Review
Week 11:
• Dec. 19 (F) 8:00–10:00 a.m. Final Exam