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A. Iarrobino, Math U575, Spring 08 Classnotes
Math U575 class notes, Spring 2008
Prof. A. Iarrobino
For Section
meeting MWTh at 10:30 AM in 178FWV (Key #03247).
Look here for specific assignments and brief comments on what we did (or will do) in class.
This will be updated at least
weekly and usually by 4:30 PM each class day. TBA = "to be announced"
Spring 08 syllabus (informaton, topics, problem list): (pdf)
First day of classes: Monday January 7 After the prequiz (to gauge preparation of class), we went over
syllabus, information, then introductions.
We worked on finding geometric symmetries of a one-simplex (straight line AB), and of a rectangle (corners labelled counterclockwise
{A,B,C,D}, from bottom left. We used several notations, including
cycle notation (AB)(CD); row notation.
We also wrote out Cayley tables for these two
automorphism groups, and noted their symmetries and patterns. [One is the cyclic group Z_2 with two elements, the other Z_2 + Z_2 (we'll see later
what this means).
Each row/column contained all elements of the group once. (true for all Cayley tables).
The two Cayley tables are symmetric about the main diagonal (transpose M^T=M). We noted
that this is from the commutativity of these groups. (ab=ba always)
The two tables were also symmetric
about the other diagonal (lower left to upper right).
The main diagonal had only e, for each group (every element has square the identity).
The products of the elements in each row/column has a constant value.
Prequiz.
(if you wish to look at it some more)
HW for Wednesday January 9:
- Worksheet 1 (handed out in class), do a,c,d, B#a,b. Note correction
in #Ba: Let S be a set with an associative operation and an identity e. Show that a right inverse A
of B in S, and a left inverse C must be the same.
Worksheet #1, revised.
- Read for Thursday in text Chapters 1,2 p.31-52.
Wednesday January 9 Worksheet problems, S_3 = D_3 (symmetries of equilateral triangle,
D_5 (symmetries of regular pentagon), Worsksheet 1 #Ba.
Further work with symmetry groups and their tables.
Subgroups of S_3, D_5, from patterns in the Cayley (group multiplication)tables
Homework for Thursday: Complete the worksheet (except make up problems).
Thursday, January 10 From Worksheet: #Bb (the inverse of AB in a group).
We continued from worksheet with examples of groups, group properties: We considered
D_4 (symmetries of square), finding all subgroups (3 of order 4, 5 of order 2, 1 of order 1).
We discussed that one order 4 subgroup (the rotations) is isomorphic to Z_4 (integers mod 4); and
the other order four subgroups had a quite different table, where each group element is its own inverse.
We also discussed a normal subgroup (the rotations) and the quotient group D_4/R = {(rotation, flip)}
Z_2.
We determined the elements of the symmetry group for infinite railroad tracks (WS #1Ae),
using the orbit of a single vertex A (where it can be moved), and the subgroup that fixes A (a flip about the
railroad tie through A).
We defined "Group": text p. 43, and gave as examples, Z,+, Q* under multiplication, Q/Z.
For more examples see p. 49 of text.
I proposed plan that students working with a few neighbors, would each research some famous group, or well known
kind of group (I will provide a list), and report back to the class. If there are groups that you
particularly wish to learn about, please let me know Monday.
Homework for Monday January 14
Worksheet #1: #1e. Please consider generators and relations for the symmetries of an infinite
railroad track. How can we express the many motions we discussed, in terms of translations,
horizontal flip about median, and one vertical flip about the tie through A? (so all other
motions that we discussed should be written in terms of these). See discussion of orbit of A, and
the subgroup of motions fixing A, above.
Chapter 1. p. 37 #5,9,12-13,17,19,21.
Chapter 2 p. 53 #5,8,11,14, 19 (please give a math induction proof).
For practice in math induction: See text p. 14-16, do Chapter 0 #11,24,28. (Please do these if
you did not get the math induction on the placement correct, or if you would like more practice).
If you missed the problems on practice quiz #B,C,G please review matrix product, and inverse, and nullspace
from a linear algebra text. Of these the product and criterion for when the inverse exists is most important.
Week of January 14 Examples of groups, properties of groups, subgrouprs - centralizer as
subgroup. Cyclic groups, intro to permutation groups (first time through). Text chapters 1-4, some of 5.
Actions of a group on a set. Quiz #1
Thursday (see coverage below)
Monday January 14 snow day, NU closed.
Wednesday January 16 Checked HW. Discussion of chapter 2 p. 53. #5 (Nguyenho),14 (Kyle),19
(Diane) (math induction).
[On #5 (inverse of a matrix over Z_{11}), Nguyenho used a standard formula, then used inverses in
the multiplicative group of Z_{11}.
The usual row reduction process does work.
[On #24 we added to Kyle's proof, a step of, given (a,b) in G, we identified an element c such that
ab=ca, and then fed this triple c into the assumption, as Kyle did.
[On the math induction, we somewhat reorderd steps, to conform to
Show P(1)
Assume P(n) (write it out)
Show that it implies P(n+1) (this is the main part).
State that P(n+1) has been shown
Then boiler plate: ``By Math induction, P(n) is true for all n".
Homework for Thursday (and Wednesday Jan 23).
Read Chapter 5, p. 94-99 (cycle decomp): do p. 112 #1, and #4 (give cycle decomp).
Read Chapter 3. Do p. 67 #2,3,10,14,15,16.
Chapter 2: choose a selection from #6,20,22-23,25,32,34,36-37
Thursday January 17 Some discussion of Chap 2 p. 56 #37 (2x2 matrices, entry all same).
Quiz 1
Homework for Wednesday Jan 23
See that just above before Thursday Jan 17 entry. Also
Chapter 3. p/ 67 #18,20,21,28,29,51.
Read Chapter 4 (cyclic groups). [If time do p. 82 #1,3.]
Group topics for report Please come in Wednesday Jan 23 with a short (several sentence outline or description)
for each of two possible topics
(could be particular groups, or aspects of groups, or an application),
about which you might be interested to
report back to the class (written or/and oral).
There are resource links on this page, and also in the text.
The intent is that particular topics could be reported on by 2-4 students: we'll discuss how to set this up.
If you know others interested in similar topics that could be helpful.
Note I will be at a conference in Halifax, and out of e-mail contact 1/17 4 PM to Tuesday 1/22 6 AM)
Week of January 21 Group properties, subgroups continued. Cyclic groups (chapters 4).
Also, actions of a group on a set, direct product (or sum) of
groups (definitions, so we can use them).
Monday January 21 Martin Luther King Holiday, no classes.
Wednesday January 23 Quiz 1 back. went over.
Questions on HW chap 3. Issues of infinite
intersection vs finite intersection as Chap 3. p. 68 #14,15.
.
Thursday January 24. Questions. Chap 3 #51 (centralizer, center).
Chapter 2 #37 group
of matrices with determinant zero, under mult., not a subgroup of Gl(2).
Cyclic groups Z, Z_n. Order of an element, of
a group, the cyclic group generated by $a^k$ in $Z_n$ (multiplicative),
mult vs additive notation.
Theorem 4.2, illustrated by $n=24, k=3, then k=15$.
Relevance to Fund Thm of Abelian Grps.
Collected first topic outlines.
Homework for January 28 Syllabus HW, chapter 4
Week of January 28 Cyclic groups, permuation groups, choose topics. Quiz 2 Thursday.
Monday January 28 Questions on HW (cyclic groups) #27,28
HW for Wednesday, January 30
(if time) Read Chap 5 thruogh p. 94- 105, begin syllabus HW.
Optional; Supplementary problems chap 4,
Do some.
p. 90 #1-4,5, 6 (Hiint, use #5, show ab is conjugate to ba)), 8 (this is like a puzzle), 10,16,17*,30,33.
(Feel free to replace some of assigned from Chap 4
with some of these).
Wednesday January 30 Questions on cyclic groups,p. 82 #21,49,63. Also, we discussed
generators of <9> subset (Z_24}, Used that <9> in @_{24} is isom to Z_8=Z_{24/GCD (24,9)).
Isomorphism: same group structure, maybe different names for elements.
Symmetric groups: cycle, cycle decomposition, order of cycles, disjoint cycles commute,
order of $S_n$, $A_n$ alternating group.
Homework for Thursday Jan. 31
HW that seems reasonable from what we know (after today's class and reading through p. 105):
Chap 5 p. 112: #1-9,17,18,23,24,28.
Hint #8: very close to #49 p. 85 (chap 4) discussed in class today.
Note that disjoint cycles commute.
Thursday January 31
We went over cycle deomposition and order, inverses, of permutations (chap 6).
Also, writing a
permutation as product of transpositions. Quiz 2.
HW for Monday, Feb 4 Please catch up, if needed, so you are ready to begin Chapter 6.
Complete the syllabus problems from Chapter 5
Chapter 1-4 supplementary problems p. 90 #1-10, 17.
Look over report topics, and choose by Monday or Wednesday a topic to outline in more detail.
(if time) Read p. 122-128 in Chapter 6, do p. 132 #1
Group theory project topics handed in by
class (one from me).
Week of February 4 Isomorphism of groups (chapter 6), multiplicative gropu U(n). Product group
Monday February 4. Check HW, questions. Discuss feedback.
Isomorphism of groups. Homomorphism.
Example Z to 2Z. Example of homomorphism: Q to Q+Q(sqrt(2), by a to a+a(sqrt (2) (not isom as not onto).
Wednesday February 6 Isomorphism, continued, U(n) for n=4,5,6,7,8,10,12. (chap 6); finding a generator
when U(n) is cyclic (as 3 or 5 when n=7.
Product G\oplus H of groups (p. 153 of text). Example: Z_2+Z_3, we showed
it is isomorphic to Z_6, and has generators (1,2) and (1,1).
Discussed prerferences for projects.
Homework for Thursday
Continue Chap 6 HW from syllabus.
(Optional) Chap 8 #2,4,6>.
Thursday February 7 Chapter 5 #6,7 (orders of subgroups of A_n, S_n).
Isomorphisms #25 (Q not isom to Z).
Lemma p 102 on product of transpositions equal identity (Hannah and Jared)
The proof uses complete induction.
Cayley Theorem 6.1: every group is isomorphic to a subgroup of a permutation group (Christian and Siobhan).
Note that a group G of order n is isomorphic to hat (G) = { T_g, g\in G}, a subgroup of Sym (G): Sym (G) = the
symmetric group on the set {G}. Note Sym G is a huge group, of order n!. Thus, Cayley's method embeds Z_6
via the ``left regular representation" into
the permutation group S_6 of order 6!=240.
One might ask, what is the smallest permutation group in which Z_6 embeds?
In which a given group G embeds?
Inner automorphism vs automorphism. Note, for G Abelian, In(G)=G.
Homework for Monday February 11
Chapter 8 #2-6,8,9.
Prepare one of Lemma p. 102, or Theorem 6.1 to outline steps/prove on a quiz.
A. Find the smallest k for which Z_6 embeds in S_k. Same question for Z_{12}=Z_4 + Z_3.
B. Let g be a 5 cycle and g' a three cycle in S_n, then the group H=Z_5+Z_3 (product) embeds in A_n,
by Z_5+Z_3 = {(g)+(g')} goes
to the subgroup (g,g') generated by g,g' in S_n. Also show that H is commutative.
(warning: this is NOT a Cayley theorem embedding).
C. Similar question for the group K generated by two disjoint three cycles g, g' in A_n.
Week of February 11 Presentations from chapter 6. Intro to product of groups, continued.
Begin chapter 7: cosets, Lagrange theorem. Quiz 3 Wednesday or Thursday
Monday February 11 Presentations from chapter 6. Product of two groups, continued.
Cosets of a subgroup.
Wednesday February 13 Worksheet 2 #1,2 (start). Present Thm 6.2. Begin cosets.
Thursday Feb. 14. Questions, Cosets. Quiz 3
Quiz 3 solutions
Homework for Wed Feb 20 and Thurs Feb 21:
a. Study solutions to quiz 3. I'd like students to understand these well, and do well on a
related half quiz Monday Feb 25.
b. Complete Worksheet #2, #1-4 (including comment on the rewrites).
c. Read chapter 7, Do syllabus HW up to #18.
Week of February 18 We focused further and more intensively on Chapter 6, the material of quiz 3, and we began Chapter 7.
Monday February 18. President's Holiday
Wednesday February 20 Quiz 3 back.
Thursday February 21 Continue Chap 6,7 intensive. Collected HW Chap 7. Presentations, Theorem 7.2 (groups of order 2p)
Week of February 25 Quiz 3.5 Monday. Chapter 7, continued. Stability group, orbits.
Significance of
left and right cosets of Stab_G(a). Conjugate subgroup, normal subgroup.
Monday February 25 Questions on Chapter 6,7. Quiz 3.5 (see below)
HW for Wednesday. Complete syllabus HW, chapter 7
Wednesday February 27 Quiz 3.5 back, go over #1. Stability groups, orbits. Example 9 (soccer ball),
Example D_4 applied to i) corners of square ii. interior and boundary of square (find stabilizer of
an element and its orbit. D_4 applied to the diagonals of a square.
Thursday February 28 Stabilizer of {4} in action of S_4 on {1,2,3,4}, as subgroup H isomorphic
to S_3 acting on {1,2,3} (H leaves 4 fixed).
We showed that each left coset (4,3) H is all group elements taking 4 to 3.
The right coset H(4,3) is all group elements taking 3 to 4. Proof of this.
Conjugate subgroup gHg^{-1} of a subgroup H of G.
Thm gHg^{-1} is isomorphic to H.
Normal subgroup (left cosets are right cosets). Equivalently, H normal if and only if every conjugate
of H in G is (identical to) H.
Any subgroup of an Abelian group is normal.
We noted the table 6.1 p. 121 as giving form of homomorphisms.
Homework for Monday March 10
Write half page of commentary on the text, in form that can be sent by e-mail to
David.
Please print it out for me to look over Monday when we are back.
Read Chapter 8 through p. 159 (rest optional). Do syllabus HW (#1-20, 49). Questions
bring to class. (i'm thinking that since we've discussed products of two groups, students will
be able to read this topic themselves and do problems. Let's see.
Chapter 9 #1,3,4-6,8 only (normal subgroups, no factor groups yet).
Week of March 3: Have a good vacation! (rest and catch-up!).
Week of March 10 Products (Chapter 8), Normal subgroups.
Quiz 4 Thurs.
Monday, Wednesday, March 10, 12 Questions on problems Chapter 8. Definition of normal subgroups,
verifying if a subgroup is normal. On Wednesday we went over an extended
example of Z_{25}+Z_{25}, counting elements of order 25, and intersection of two cyclic subgroups
of order 25.
Also, proof that if a subgroup H of G has index 2, then it is normal in G (problem #7, chap 9).
Thursday, March 13.. Quiz 4.
Math U575 Quiz 4, Spring 08 (pdf)
Solutions to Math U575 Quiz 4, Spring 08 (pdf) (corrected, 1.4MB)
Homework for Monday, March 17
Supplementary Problems after Chapter 8, p. 174. #5,21,47,50.
Same set, choose about 4 from 6-8, 13-14,16-17,22,25-26.
Read Chapter 9, p. 179-181, in preparation for Monday's class.
(a first read, to get an impression. I will present it, and we will go over it quite a bit).
Monday March 17 Pass back Quiz 4 and go over questions on it. Pass back solutions
Give out Preparation, sample problems for Exam 1.
Factor (quotient) groups, and relation with normal subgroups, and kernel of homormorphism.
Example H=5Z in Z,+, with quotient Z_5.
Example A_3 in S_3, with quotient isomorphic to Z_2,+.
Homework for Wednesday March 19
Chapter 9, #10, 12.
See HW for Monday above
Wednesday March 19 Factor groups, relation with group homomorphisms, continued
Thursday March 20 Worksheet #3 - practice problems for Exam 1 (solved in groups, students put solutions
on board.
Worksheet 3, Math U375
We went over the hint in Problem 2 (looking at H as the stability (same as isotropy) group of an element P_1={AB,CD} of the set P of
pairs of opposite edges of the tetrahedron. That H is also the isotropy of each element of the orbit of P_1 (each element of P)
shows that H is normal in G, the group of automorphisms of the tetrahedron.
Relevant to #4b, we noted that for an order 2 subgroup H=(g,e) of G to be normal, g must be in the center of G, i.e. commute with each
element of G (special for order 2 subgroups, not true for H of order greater than 2).
Correction to passed out solution sheet of Quiz 4, thanks to Kyle (the solution sheet posted above is already corrected)
#3b: .... or else has projection on the first component 0 or 7Z. ...
#5v. SOL (calculate) (ACD) {(ABC),(ACB),e)}={(ABD),(AD)(BC),(ACD)} (these take the face F_1 to F_4).
Now {(ABD),(AD)(BC),(ADC)}(ADC)={(BDC),(BCD),e},
which is the stabilizer of F_4, the face BCD of the tetrahedron.
Week of March 24 Exam 1 Monday, Group homomorphism, continued, Structure theorem of finite Abelian groups,
Monday, March 24 Exam (full class period)
Homework for Wednesday
Read Chapter 10 up to page 205.
Do p. 210 (Chapter 10) #5, #8 for S_4, 9.
Wednesday March 26 Structure of finite Abelian groups
Thursday March 27 Exam 1 back. Quiz 5.6 of Prof. Todorov (Quiz 6 handed out). Topics: group actions
on a set.
HW for Monday March 31 Complete Chap 10 #5-22, 48,53.
Complete quiz 6 of Prof. Todorov section.
Week of March 31 Structure of finite Abelian groups (chap 11). Symmetry and counting (chap 29 and worksheet 4).
Begin Sylow Theorems (Chapter 24).
Monday March 31 We discussed the structure of finite abelian groups
(chapter 11), and then the Burnside Theorem (chap 29).
HW for Wednesday
Chapter 11, #1-6.
Chapter 29 #1-3.
Worksheet #4 page 1 (read, and answer Question 1)
Worksheet 4, page 1 (Burnside theorem)
Note Please send your updated comments on the text to
largent.d@neu.edu
as attachments, with header: Text comments.
Please state "sign" or "do not sign" at the beginning of the message (opt out).
Wednesday, April 2. Burnside theorem continued, coloring a tetrahedron (worksheet 4 p. 2).
Begin Sylow theorem.
Week of April 7 Sylow Theorems, handout on Sylow applications, Quiz 5 Wednesday,
Solutions to Quiz 5 Thursday, Prof. Todorov Quiz 8.
HW for Monday April 14 Quiz 7 and 8 of Prof. Todorov (pass in Monday); first side of
Sylow Applications (check Monday).
Week of April 14 Sylow applications, review. Presentations of projects (either to me in
office or in class).
All projects due Wednesday April 15 at 5 PM Presented to me orally at least.
Forthcoming featured events
Final Exam Tuesday April 22 at 10:30 PM: Two hour exam
Be sure to make
your travel plans so you can take the final, required of all. Bring graphing calculator
and Extra Credit HW (see above).
Archived events
Quiz 5 Finite Abel groups, Burnside Thm, Sylow thms: April 9.
Exam (hour) Monday, March 24
Chapters 1-9, emphasis on 5-8, and normal subgroup. Factor groups as extra credit. Format: 4 problems (25 points each), chosen out of 5; also 2 extra credit problems (10 pts each)
See sample problems for Exam 1 Notes for Exam 1.
Also #1 (cosets) on Quiz 4 (cosets) (See Quiz 4 and corrected solutions posted elsewhere on page).
Also
see Worksheet 3, Math U375 (practice for Exam 1).
Quiz 4 Wed. March 12 or Thursday March 13. Chaps 7-9. Group actions, Stab_G(a), Orbit_G(a), significance of left, right cosets of Stab_G(a) in G,
conjugacy, normal subgroups.
Math U575 Quiz 4, Spring 08 (pdf)
Solutions to Math U575 Quiz 4, Spring 08 (pdf) (corrected, 1.4MB)
Quiz 3.5 Monday Feb. 25.
Quiz 3.5
Quiz 3 Thursday, Feb 14: Chapters 5,6, Thms presented. Product of groups. Cosets.
Quiz 2: Thursday Jan 31:
Quiz: main emphasis: Chapter 4 (cyclic groups), HW and class work.
Quiz #1 Thursday January 17: Worksheet #1, Text Chaps 1-2, notation from chapter 5,
discussions in class.
Also math induction, matrix multiplication and inverse. (45 minutes).
In chapter 3 you might review notes of problems discussed in class; #4,15, 51; also #16.
Reminder: attendance policy The attendance policy (announced in Syllabus: information sheet):
Briefly,
more than four unexcused absences affects your grade, or may result in a W or F in the course.
Questions or comments: e-mail to a.iarrobino@neu.edu. This is the quickest way to reach me, or come by 526 NI, MWTh.
Links to group resources on the web:
New York Group Theory cooperative Excellent, authoritative source for group theory, including open problems. See links to ``Magnus'' open source software, and to ``Projects and problems"
GAP
(groups, algorithms, programming system [Several Northeastern CS profs as Gene Cooperman are developers of GAP]
Many Mathematics Department Professors use groups a great deal in their research: Profs. Donald King (nilpotent orbits, Lie groups), Lakshmibai (Schubert calculus, matrix groups), , Valerio Laredo-Toledano (quantum groups), Marc Levine (K-theory), David Massey (intersection homology)
Egon Schulte (geometric combinatorics, polytopes),Alex Suciu (homotopy and homology groups, hyperplane arrangements, Chen groups), Gordana Todorov (Artin algebras), Jerzy Weyman (nilpotent orbits, algebraic resolutions), Andrei Zelevinsky
(Lie groups, combinatorics), and myself (Artin algebras, nilpotent matrices),
you can see
references to this use on some of our websites.
Basic pictorial intro to symmetries, wallpaper tilings-``Science U geometry center"
Some basic concepts and links - Tom Banchoff site at Brown U."
Wallpaper groups (David Joyce site at Clark)
General intro to groups, and applications from Springer Encyclopedia of Mathematics (online)
Prof. Anthony Iarrobino
Department of Mathematics
Northeastern University
Boston,
MA,
02115
Office:526 NI
Phone: (617) 373-5524
Email: a.iarrobino@neu.edu
Created: January 6, 2008. Last modified: April 10, 2008.
URL:http://www.math.neu.edu/~iarrobino/AIMathU575Spr08classnotes.html