Welcome to Math 1100! (additions to this web site no longer count towards good deed points)

#

Week of...

Notes and Links

1

Sep 8

About This Class; Monday  Non Commutative Gaussian Elimination; Thursday  the category of groups, automorphisms and conjugations, images and kernels.

2

Sep 15

Monday  coset spaces, isomorphism theorems; Thursday  simple groups, JordanHolder decomposition series.

3

Sep 22

Monday  alternating groups, group actions, The Simplicity of the Alternating Groups, HW1, HW 1 Solutions, Class Photo; Thursday  group actions, OrbitStabilizer Thm, Class Equation.

4

Sep 29

Monday  Cauchy's Thm, Sylow 1; Thursday  Sylow 2.

5

Oct 6

Monday  Sylow 3, semidirect products, braids; HW2; HW 2 Solutions; Thursday  braids, groups of order 12, Braids

6

Oct 13

No class Monday (Thanksgiving); Thursday  groups of order 12 cont'd.

7

Oct 20

Term Test; Term Test Solutions on Monday, HW3; HW 3 Solutions; Thursday  solvable groups, rings: defn's & examples.

8

Oct 27

Monday  functors, CayleyHamilton Thm, ideals, iso thm 1; Thursday  iso thms 24, integral domains, maximal ideals, One Theorem, Three Corollaries, Five Weeks

9

Nov 3

Monday  prime ideals, primes & irreducibles, UFD's, Euc.Domain$\Rightarrow$PID, Thursday  Noetherian rings, PID$\Rightarrow$UFD, Euclidean Algorithm, modules: defn & examples, HW4, HW 4 Solutions

10

Nov 10

Monday  R is a PID iff R has a DH norm, Rmodules, direct sums, every f.g. module is given by a presentation matrix, Thursday  row & column reductions plus, existence part of Thm 1 in 1t3c5w handout.

11

Nov 17

MondayTuesday is UofT's Fall Break, HW5, Thursday  1t3c5w handout cont'd, JCF Tricks & Programs handout

12

Nov 24

Monday  JCF Tricks & Programs cont'd, tensor products, Thursday  tensor products cont'd

13

Dec 1

EndofCourse Schedule; Monday  tensor products finale, extension/reduction of scalars, uniqueness part of Thm 1 in 1t3c5w, localization & fields of fractions; Wednesday is a "makeup Monday"!; Notes for Studying for the Final Exam Glossary of terms

F

Dec 15

The Final Exam

Register of Good Deeds

Add your name / see who's in!

See Non Commutative Gaussian Elimination


Bad news. This assignment is due at class time on Thursday, October 16, 2010.
Good news. It will be marked by the following day, Friday October 17, and be available for pickup at my office between 24PM, providing you timely feedback for the term test on the following Monday.
Solve the following problems
(but submit only your solutions of problems 2, 3, 4, 8, 9, and 11).
Problem 1. (Selick)
 What it the least integer $n$ for which the symmetric group $S_{n}$ contains an element of order 18?
 What is the maximal order of an element in $S_{26}$? (That is, of a shuffling of the red cards within a deck of cards?)
Problem 2. (Selick) Let $H$ be a subgroup of index 2 in a group $G$. Show that $H$ is normal in $G$.
Problem 3. Let $\sigma \in S_{20}$ be a permutation whose cycle decomposition consists of one 5cycle, two 3cycles, and one 2cycle. What is the order of the centralizer $C_{S_{20}}(\sigma )$ of $\sigma$?
Problem 4. (Selick) Let $G$ be a group of odd order. Show that $x$ is not conjugate to $x^{1}$ unless $x=e$.
Problem 5. (Dummit and Foote) Show that if $G/Z(G)$ is cyclic then $G$ is Abelian.
Problem 6. (Lang) Prove that if the group of automorphisms of a group $G$ is cyclic, then $G$ is Abelian.
Problem 7. (Lang)
 Let $G$ be a group and let $H$ be a subgroup of finite index. Prove that there is a normal subgroup $N$ of $G$, contained in $H$, so that $(G:N)$ is also finite. (Hint: Let $(G:H)=n$ and find a morphism $G\to S_{n}$ whose kernel is contained in $H$.)
 Let $G$ be a group and $H_{1}$ and $H_{2}$ be subgroups of $G$. Suppose $(G:H_{1})<\infty$ and $(G:H_{2})<\infty$. Show that $(G:H_{1}\cap H_{2})<\infty$
Problem 8. (Selick) Show that any group of order 56 has a normal Sylow$p$ subgroup, for some prime $p$ dividing 56.
Problem 9. (Qualifying exam, May 1997) Let $S_{5}$ act on $({\mathbb {Z} /5})^{5}$ by permuting the factors, and let $G$ be the semidirect product of $S_{5}$ and $({\mathbb {Z} /5})^{5}$.
 What is the order of $G$?
 How many Sylow5 subgroups does $G$ have? Write down one of them.
Problem 10. (Selick) Show that the group $Q$ of unit quaternions ($\{\pm 1,\pm i,\pm j,\pm k\}$, subject to $i^{2}=j^{2}=k^{2}=1\in Z(Q)$ and $ij=k$) is not a semidirect product of two of its proper subgroups.
Problem 11. (Qualifying exam, September 2008) Let $G$ be a finite group and $p$ be a prime. Show that if $H$ is a $p$subgroup of $G$, then $(N_{G}(H):H)$ is congruent to $(G:H)$ mod $p$. You may wish to study the action of $H$ on $G/H$ by multiplication on the left.