Course webpage for MA 541 (Modern Algebra I), Fall 2019

Archived version

Instructor: Anna Medvedovsky
Course syllabus

Homework assignments

References are to the course text: Fraleigh, A First Course in Abstract Algebra, 7th ed. (F)
Homework deadline: 4:45pm on the due date in the 541 "dropbox" in the main office (MCS 123).

Week In-class topics Homework assignment Due date
1 Groups: definitions and examples (F secs. 2, 4)
Symmetries of an equilateral triangle (F example 8.7)
Groups of order 1, 2, 3
Read F section 2.
Problems: F 2.8, 2.10

Read F section 4.
Problems: F 4.2, 4.5, 4.8, 4.12, 4.18, 4.19, 4.20 (just the prompt
in the paragraph before parts a, b, c), 4.35

Optional challenge problem #1 (may be turned in 9/12/19 in class)
9/10/19
2 Subgroups (F sec. 5)
Equivalence relations (F sec. 0)
Properties of functions (F sec. 0)

Homework assignment #2

Optional challenge problems #2 (due 9/19/19 in class)
9/17/19
3 Homomorphisms of groups and examples (F sec. 13)
Isomorphisms of groups (F sec. 3)
Number theory lemmas: division algorithm (F sec. 6)

Homework assignment #3

Optional challenge problems #3 (due 9/26/19 in class)
9/24/19
4 Number theory: GCDs, Bézout's lemma (F sec. 6)
Euclid's algorithm, units modulo n
Cyclic groups (F sec. 6)

Homework assignment #4

Optional challenge problems #4 (due 10/3/19 in class)
10/1/19
5 Well-definedness
Quiz
Generators and relations
(F sec. 7 through Thm 7.6; see also F sec. 40)
D4: Symmetries of a square (F example 8.10)

Homework assignment #5 (optional challenge parts sprinkled in)

10/8/19
6 Subgroups of D4 (F example 8.10)
Permutation groups (F sec. 8)
Sn for n = 1, 2, 3, 4
Orbits, cycle notation, cycle structure (F sec. 9)
S4 in detail, including three copies of D4 inside
Sign of a permutation... (to be continued)

Homework assignment #6 (no challenge problems this week)

10/22/19
at the start of class
7 Sign of a permutation (F sec. 9)
An as a subgroup of Sn (F sec. 9)
A4 and rotations of a regular tetrahedron

No new homework this week.
8 More on the parity of a permutation, determinants
Cayley's theorem (F sec. 8) and examples
Cosets and examples (F sec. 10)

No homework. Take-home midterm due 9:30am October 29.

9 More cosets and examples (F sec. 10)
Lagrange's theorem (F sec. 10)
Consequences of Lagrange's theorem:
Fermat's Little Theorem
Only one group of order p (F Theorem 10.11)
Only two groups of order 2p

Homework assignment #7

11/5/19
10 The dihedral group Dn (F exercise 8.44)
Groups of order 2p redux
When do left cosets form a group? (F Theorem 14.4)
Normal subgroups (F definition 13.19)
Quotient groups / factor groups (F Theorem 14.5)
Fibers of group homomorphisms (F Theorem 13.15)
Kernels are normal (F Corollary 13.20)

Homework assignment #8

Special set on the cyclicity of Zn×

11/12/19
11 Bijection between fibers and image
First Isomorphism Theorem (F Thm 14.11, F Thm 34.2)
The surjection S4S3 geometrically on a tetrahedron
Universal property of quotient groups
Conjugation and inner automorphisms (F sec. 14)
("conjugacy class" defined in F sec. 37)
Homomorphisms out of groups given by presentations
(see F sec. 39 for free groups
and F sec. 40 for group presentations)
Third Isomorphism Theorem (F Theorem 34.7)
Do N and G/N determine G? (see also F. sec 35)

Homework assignment #9

11/19/19
12 Simple groups (F sec. 15)
A5 is simple (see also F exercise 15.39)
Composition series, composition factors (F sec. 35)
Jordan-Hölder theorem statement (F Theorem 35.15)
Classification of finite simple groups (informal)

Group actions: defintions and examples (F sec. 16)
G → Perm(X) perspective (F Theorem 16.3)
faithful actions, transitive actions
orbits, stabilizers
Fundamental theorem of group actions (statement)
orbits partition (F Theorem 16.14)
stabilizers are groups (F. Theorem 16.12)
orbit-stabilizer connection (F. Theorem 16.15)
[Additional reference for group actions:
Sections 1, 2, 3 of Keith Conrad's notes]

Homework assignment #10 (minor edit 11/21/2019)

11/26/19
12/03/19 in class
13 Review of definitions for group actions
new terms: fixed point, kernel of the action
Proof of orbit-stabilizer theorem (F Thm 16.15)
Illustration of theorem:
Symmetries of tetrahedron acting on vertices,
edges, center, pairs of opposite edges
Theorem: if a group of order pn acts on a set X, then
|X| ≡ |{fixed points}| modulo p (F Thm 36.1)
A group G of even order has an element of order 2
(let Z2 act on G by inversion; apply theorem)
Statement of Cauchy's theorem (F Thm 36.3)

Thanksgiving break -- no new HW.

But here's a doozy to think about: Consider the group of symmetries of a regular tetrahedron, as in class, acting on all the points of the tetrahedron. What does the orbit-stabilizer theorem look like for a randomly chosen point along an edge (that is, not a vertex and not a midpoint of the edge)? For a randomly chosen point on a median of one of the faces (that is, not on any edge and not in the center of the face)? [A median of a triangle is a line through a vertex and the midpoint of the opposite side.] For a random point on one of the faces (not on an edge or on any of the medians)? For various points in the interior of the tetrahedron? Is there a point with trivial stabilizer? For which d dividing 24 is there a point with orbit of length d?

14 Proof of Cauchy's theorem (F Thm 36.3)
Groups of order pq are never simple:
Cauchy's theorem guarantees subgroup of order q,
which, if p < q, is normal by 3b on HW #10.
Group acting on itself by conjugation
orbits are conjugacy classes
stabilizers are centralizers
Class equation (F def. 37.2)
Center of a p-group is nontrival (F def. 36.2, Thm 37.4)
Groups of order pn with n ≥ 2 are never simple
p-Sylow subgroups (F def. 36.9: a priori differs from the
definition in class, but Sylow I reconciles the two)
Sylow theorems statement (F Thms 36.8, 36.10, 36.11)
No simple groups of order 12:
Sylow III ⇒ n3 = 1 or 4. If n3 = 4, group has 8 elts
of order 3, so only space for one 2-Sylow.

p-Sylow subgroup analysis for groups of order 12:
Z12Z2 × Z6D6A4
(there's one more group of order 12:
a semidirect product of Z4 acting on a normal Z3)
No simple groups of order...
18 (the 3-Sylow is normal)
20 (the 5-Sylow is normal)
24 (either 2-Sylow is normal, or conjugation
permutes 3 of them yielding map GS3)
Fundamental theorem of finitely generated abelian groups
statement (F Thm 11.12)
example: abelian groups of order 72
hint of proof (see F Thm 38.12)

Homework assignment #11
(12/5/2019: minor clarification for optional challenge problem.)
(12/9/2019: note on what is meant by "cube" in problem 2,
nontriviality condition n > 1 added to problem 4.)

12/11/19
(Wednesday!)
15 Five platonic solids
Rotational symmetries of the cube (F exercise 12.42)
Rotational symmetries of the icosahedron/dodecahedron
Classification of finite rotation groups

[Additional referencs for this topic:
• Goodman, Algebra: abstract and concrete,
sections 4.1, 4.2, 10.3, and appendix F
• Artin, Algebra, section 5.9]
None!
Final exam 9-11am Tuesday, December 17, in MCS B31.

Sundries

Mini-film on Hagoromo chalk (3.5min, from Great Big Story)
Hagoromo chalk full documentary (25min, in Japanese)

Subgroup diagram of S4 (Susan Goldstine)

Sample subgroup diagrams using tikz: LaTeX file, pdf.
Sample matrices: LaTeX file, pdf.