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Week of...

Notes and Links

Additions to the MAT 1100 web site no longer count towards good deed points

1

Sep 12

About This Class, Tuesday  Non Commutative Gaussian Elimination, Thursday  NCGE completed, the category of groups, images and kernels.

2

Sep 19

I'll be in Strasbourg, class was be taught by Paul Selick. See my summary.

3

Sep 26

Class Photo, HW1, HW1 Submissions, HW 1 Solutions

4

Oct 3

The Simplicity of the Alternating Groups

5

Oct 10

HW2, HW 2 Solutions

6

Oct 17

Groups of Order 60 and 84

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Oct 24

Extra office hours: Monday 10:3012:30 (Dror), 5PM7PM @ Huron 1028 (Stephen). Term Test on Tuesday. Summary for the midterm exam. Term Test  Sample Solutions

8

Oct 31

HW3, HW 3 Solutions

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Nov 7

MondayTuesday is November Break, One Theorem, Two Corollaries, Four Weeks

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Nov 14

HW4, HW 4 Solutions

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Nov 21


12

Nov 28

HW5 and last week's schedule

13

Dec 5

Tuesday  the Jordan form; UofT Fall Semester ends Wednesday; our Final Exam took place on Friday

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See Non Commutative Gaussian Elimination


This assignment is due at class time on Thursday, October 20, 2010.
Solve the following questions
 (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?)
 (Selick) Let $H$ be a subgroup of index 2 in a group $G$. Show that $H$ is normal in $G$.
 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 $\sigma$?
 (Selick) Let $G$ be a group of odd order. Show that $x$ is not conjugate to $x^{1}$ unless $x=e$.
 (Dummit and Foote) Show that if $G/Z(G)$ is cyclic then $G$ is Abelian.
 (Lang) Prove that if the group of automorphisms of a group $G$ is cyclic, then $G$ is Abelian.
 (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 $(G:H_{1}\cap H_{2})<\infty$