Week 
Inclass 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 
Welldefinedness
Quiz
Generators and relations
(F sec. 7 through Thm 7.6; see also F sec. 40)
D_{4}: Symmetries of a square (F example 8.10)

Homework assignment #5 (optional challenge parts sprinkled in)

10/8/19 
6 
Subgroups of D_{4} (F example 8.10)
Permutation groups (F sec. 8)
S_{n} for n = 1, 2, 3, 4
Orbits, cycle notation, cycle structure (F sec. 9)
S_{4} in detail, including three copies of D_{4} 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)
A_{n} as a subgroup of S_{n} (F sec. 9)
A_{4} 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. Takehome 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 D_{n} (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 Z_{n}^{×}

11/12/19 
11 
Bijection between fibers and image
First Isomorphism Theorem (F Thm 14.11, F Thm 34.2)
The surjection S_{4} → S_{3} 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)
A_{5} is simple (see also F exercise 15.39)
Composition series, composition factors (F sec. 35)
JordanHölder 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)
orbitstabilizer 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 orbitstabilizer 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 p^{n} 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 Z_{2} 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 orbitstabilizer 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 pgroup is nontrival (F def. 36.2, Thm 37.4)
Groups of order p^{n} with n ≥ 2 are never simple
pSylow 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 ⇒ n_{3} = 1 or 4.
If n_{3} = 4, group has 8 elts
of order 3, so only space for one 2Sylow.
pSylow subgroup analysis for groups of order 12:
Z_{12}, Z_{2} × Z_{6}, D_{6}, A_{4}
(there's one more group of order 12:
a semidirect product of Z_{4} acting on a normal Z_{3})
No simple groups of order...
18 (the 3Sylow is normal)
20 (the 5Sylow is normal)
24 (either 2Sylow is normal, or conjugation permutes 3 of them
yielding map G → S_{3})
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
(12/4/2019: added reading from F.)
(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 911am Tuesday, December 17, in MCS B31.

