Math 71 - Abstract Algebra - Dartmouth College

General Info | Day-to-day

Learning TeX: You'll need to install an editor (my favorite for Macs is TeXShop, and I've heard TeXMaker is good for other platforms). You'll also need a TeX distribution; everything is linked to from the math department's page on TeX. A good overall guide to learning LaTeX is the Not So Short Guide. I use TikZ for pictures; the manual I use for this is Till Tantau's PGF Manual, but really I learned by digging around the TikZ example database (in general, TeXample.net is awesome). I also have a samle TeX file on my teaching page

Week 0 Read before class: 0.1, 0.2, 0.3 Basic properties of sets, integers, and modular arithmetic.
Week 1 1.1, 1.2, 1.3: Definition of a group and examples. Dihedral groups, symmetric groups.
Slides: [Monday] [Wednesday] [Friday]
Due 9/14: Homework 1 in pdf or tex
Week 2 1.5, 1.6, 1.7, 2.1, 2.2: group homomorphisms and actions, subgroups, quaternions, special subgroups
Slides: [Monday] [Wednesday] [Friday]
Due 9/21: Homework 2 in pdf or tex

Quiz: Sections 0.1-1.6. Thursday during X-hour. [solutions]
Week 3 2.3, 2.4, 3.1: Cyclic and finitely generated groups. Quotient Groups.
Slides: [Monday] [Wednesday] [Friday]
Due 9/28: 1.7 #6, 8 (read and think about 9 and 10 for contrast), 14, 15, 17, 18, 21, 23. 2.2 #5(a), 2, 6, (read and think about 7), 9 (you proved 2.1.10(a) on the last homework), 12 (a)-(d) (read 13). 2.3 #1, 9, 11, 12, 15 (you can use 12), 19, 24.

Week 4 3.1, 3.2, 3.3: Quotient groups, orders, Lagrange, isomorphism theorems
Slides: [Monday] [Wednesday] [Friday]
Due 10/5: 2.4 #3, 14, 3.1# 5, 7, 14 (see 2.1.6 for the definition of torsion), 21, 24, 35 (recall that determinant is a homomorphism), 36, 40, 41 (read 42).

Week 5 3.5, 4.1, 4.2, 4.3: Alternating group and group actions revisited
Slides: [Monday] [Wednesday]
Due 10/12: 3.2# 4, 11, 14 (using ideas developed in 3.2; do not use brute force), 18; 3.3# 2, 3, 7, 8, 3.5#12, 17 (read 15 and 16 for help); 4.1 # 2, 7 [ hint
Recall that the elements of the orbit of $a \in A$ are in bijection with the cosets of the stabilizer $G_a$ in $G$ in a very specific way. So to show that primitive implies maximal, go the other way: assume some stabilizer is not maximal (i.e. there exists some subgroup $H$ of $G$ containing $G_a$) and show that there's some non-trivial block in $A$. To do this, consider the orbit of $a$ under the action of $H$. Show that $H$ does not act transitively or trivially (using 3.2.11 and the fact that (a) $G$ is finite, and (b) $G_a$ is also the stabilizer of $a$ in $H$). Then show that the orbit of $a$ under $H$ is a non-trivial block.
]

Quiz: Sections 1.7-4.1. Thursday during X-hour. [solutions]

Week 6 4.3, 3.4, 4.5
Slides: [Monday] [Wednesday] [Friday]
Due 10/19: 4.2 #7, 8, 10 (You may assume Cauchy's theorem), 11, 12 [ hint
Consider the alternating group inside of S_G, then use 3.3#3.
], 13, 4.3 #3, 5, 12 (just do S_8), 19 [ hint
the hint they give you helps you to calculcate k
] (read 20-22), 26 [ hint
You can show that if you conjugate a stabilizer G_a by g you get G_{ga}, so since G acts transitively, the set of elements which are in some stabilizer is the same as the union over all g in G of (g G_a g^{-1}). Use Prop 6 to count how big this union could possibly be.
], 30 [ hint
Show that if an element and its inerse are both in the same conjugacy class, then that class has even order.
].

Midterm: Chapters 1-4. Out Friday, in Wednesday. tex template
EXTRA CREDIT: Turn in, with the exam, a distilled set of notes which summarizes the material covered in chapters 1-4.
Extra office hours this week: Monday 12:30-2, Tuesday 2:30-4, or by appointment.
[Solutions: (pdf) or (tex)]
Week 7 5.1, 5.2, 7.1
Slides: [Monday] [Wednesday] [Friday]
Due 10/29: 3.4 #1, 2 (just do Q_8), 7; 4.5 #2, 3, 4 (just do S_3 \times S_3), 18, 22; 5.1 #5, 10.

Week 8 7.2, 7.3, 7.4
Slides: [Monday] [Wednesday] [Friday]
Due 11/02: 5.2 #2&3 parts (a) and (b), 5, 7 (see example 2 on p. 155), 11, 14; 7.1 (notice that all problems consider only rings with 1) #4, 6 (just give answers, no need to justify), 7, 13(a)+ give an example, 14, 15, 21 (you may assume that union and intersection are associative operations on sets), 24 just D=3, 25; 7.2 #1(c), 4 (read 3)

Quiz: Sections 5.1, 5.2, 7.1, 7.2, 7.3, and 7.4 up to prop 11. Thursday 11/8 during X-hour. [Solutions]
Week 9 7.4, 8.1, 8.2. A PID that's not a ED
Slides: [Monday] [Wednesday] [Friday]
Due 11/09: 7.2 #11, 13; 7.3 #2, 6, 10, 13, 17, 23, 26 (you may use 25), 28, 7.4 #5, 12, 16, and work through and informally write up 14. In case you've forgotten how to do long division on polynomials

Week 10 8.3, 9.1-9.4
Slides: [Monday]
Due 11/16 (full credit given for mostly completed): 7.4 # 11, 34; 8.1 #3, 4, 8: (a) D=-2, (b) D = -43; 11 8.2 #6;
Final: Chapters 0-9. Out Monday 11/12, in Monday 11/19 by 9am. [Template] [Solutions: (tex) or (pdf)] Extra office hours han be signed up for here.

Recent posts on group theory:
math.GR on arXiv

Week 0	Read before class: 0.1, 0.2, 0.3 Basic properties of sets, integers, and modular arithmetic.
Week 1	1.1, 1.2, 1.3: Definition of a group and examples. Dihedral groups, symmetric groups. Slides: [Monday] [Wednesday] [Friday]
	Due 9/14: Homework 1 in pdf or tex
Week 2	1.5, 1.6, 1.7, 2.1, 2.2: group homomorphisms and actions, subgroups, quaternions, special subgroups Slides: [Monday] [Wednesday] [Friday]
	Due 9/21: Homework 2 in pdf or tex
	Quiz: Sections 0.1-1.6. Thursday during X-hour. [solutions]
Week 3	2.3, 2.4, 3.1: Cyclic and finitely generated groups. Quotient Groups. Slides: [Monday] [Wednesday] [Friday]
	Due 9/28: 1.7 #6, 8 (read and think about 9 and 10 for contrast), 14, 15, 17, 18, 21, 23. 2.2 #5(a), 2, 6, (read and think about 7), 9 (you proved 2.1.10(a) on the last homework), 12 (a)-(d) (read 13). 2.3 #1, 9, 11, 12, 15 (you can use 12), 19, 24.
Week 4	3.1, 3.2, 3.3: Quotient groups, orders, Lagrange, isomorphism theorems Slides: [Monday] [Wednesday] [Friday]
	Due 10/5: 2.4 #3, 14, 3.1# 5, 7, 14 (see 2.1.6 for the definition of torsion), 21, 24, 35 (recall that determinant is a homomorphism), 36, 40, 41 (read 42).
Week 5	3.5, 4.1, 4.2, 4.3: Alternating group and group actions revisited Slides: [Monday] [Wednesday]
	Due 10/12: 3.2# 4, 11, 14 (using ideas developed in 3.2; do not use brute force), 18; 3.3# 2, 3, 7, 8, 3.5#12, 17 (read 15 and 16 for help); 4.1 # 2, 7 [ hint Recall that the elements of the orbit of $a \in A$ are in bijection with the cosets of the stabilizer $G_a$ in $G$ in a very specific way. So to show that primitive implies maximal, go the other way: assume some stabilizer is not maximal (i.e. there exists some subgroup $H$ of $G$ containing $G_a$) and show that there's some non-trivial block in $A$. To do this, consider the orbit of $a$ under the action of $H$. Show that $H$ does not act transitively or trivially (using 3.2.11 and the fact that (a) $G$ is finite, and (b) $G_a$ is also the stabilizer of $a$ in $H$). Then show that the orbit of $a$ under $H$ is a non-trivial block. ]
	Quiz: Sections 1.7-4.1. Thursday during X-hour. [solutions]
Week 6	4.3, 3.4, 4.5 Slides: [Monday] [Wednesday] [Friday]
	Due 10/19: 4.2 #7, 8, 10 (You may assume Cauchy's theorem), 11, 12 [ hint Consider the alternating group inside of S_G, then use 3.3#3. ], 13, 4.3 #3, 5, 12 (just do S_8), 19 [ hint the hint they give you helps you to calculcate k ] (read 20-22), 26 [ hint You can show that if you conjugate a stabilizer G_a by g you get G_{ga}, so since G acts transitively, the set of elements which are in some stabilizer is the same as the union over all g in G of (g G_a g^{-1}). Use Prop 6 to count how big this union could possibly be. ], 30 [ hint Show that if an element and its inerse are both in the same conjugacy class, then that class has even order. ].
	Midterm: Chapters 1-4. Out Friday, in Wednesday. tex template EXTRA CREDIT: Turn in, with the exam, a distilled set of notes which summarizes the material covered in chapters 1-4. Extra office hours this week: Monday 12:30-2, Tuesday 2:30-4, or by appointment. [Solutions: (pdf) or (tex)]
Week 7	5.1, 5.2, 7.1 Slides: [Monday] [Wednesday] [Friday]
	Due 10/29: 3.4 #1, 2 (just do Q_8), 7; 4.5 #2, 3, 4 (just do S_3 \times S_3), 18, 22; 5.1 #5, 10.
Week 8	7.2, 7.3, 7.4 Slides: [Monday] [Wednesday] [Friday]
	Due 11/02: 5.2 #2&3 parts (a) and (b), 5, 7 (see example 2 on p. 155), 11, 14; 7.1 (notice that all problems consider only rings with 1) #4, 6 (just give answers, no need to justify), 7, 13(a)+ give an example, 14, 15, 21 (you may assume that union and intersection are associative operations on sets), 24 just D=3, 25; 7.2 #1(c), 4 (read 3)
	Quiz: Sections 5.1, 5.2, 7.1, 7.2, 7.3, and 7.4 up to prop 11. Thursday 11/8 during X-hour. [Solutions]
Week 9	7.4, 8.1, 8.2. A PID that's not a ED Slides: [Monday] [Wednesday] [Friday]
	Due 11/09: 7.2 #11, 13; 7.3 #2, 6, 10, 13, 17, 23, 26 (you may use 25), 28, 7.4 #5, 12, 16, and work through and informally write up 14. In case you've forgotten how to do long division on polynomials
Week 10	8.3, 9.1-9.4 Slides: [Monday]
	Due 11/16 (full credit given for mostly completed): 7.4 # 11, 34; 8.1 #3, 4, 8: (a) D=-2, (b) D = -43; 11 8.2 #6;
	Final: Chapters 0-9. Out Monday 11/12, in Monday 11/19 by 9am. [Template] [Solutions: (tex) or (pdf)] Extra office hours han be signed up for here.