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

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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

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See Non Commutative Gaussian Elimination


This assignment is due at class time on Monday, October 6, 2011.
Part I
Web search "Rubik's Cube Variants" (look at images), or look at Wikipedia: Combination Puzzle or TwistyPuzzles.com, or search elsewhere or go to a toy shop, pick your favourite "permutation group puzzle" (other than the Rubik Cube, of course), and figure out how many configurations it has. For your solution to count the number of configurations must be more than you can count, and your solution must include a clear picture or diagram of the object being studied, its labeling by integers, the list of generating permutations for it, and a printout of the program you used along with screen shot of its output (or an input/output log). It is ok to use the program presented in class (Mathematica is available on a departmental server; look for it!) but better to write your own. You can submit your solution either as a wiki page on this server (best option), or as a URL elsewhere (second best), or as a single file in any reasonable format, or on paper.
Part II
Solve the following questions.
 (Selick) If $g$ is an element of a group $G$, the order $g$ of $g$ is the least positive number n for which $g^{n}=1$ (may be $\infty$). If $x,y\in G$, prove that $xy=yx$.
 (Selick) Let $G$ be a group. Show that the function $\phi :G\to G$ given by $\phi (g)=g^{2}$ is a morphism of groups if and only if $G$ is Abelian.
 (Lang, pp 75) Let $G$ be a group. For $a,b\in G$, the commutator $[a,b]$ of $a$ and $b$ is $[a,b]=aba^{1}b^{1}$. Let $G'$ be the subgroup of $G$ generated by all commutators of elements of $G$. Show that $G'$ is normal in $G$, that $G/G'$ is Abelian, and that any morphism from $G$ into an Abelian group factors through $G/G'$.
 (Lang, pp 75) Let $G$ be a group. An automorphism of $G$ is an invertible group morphism $G\to G$. An inner automorphism is an automorphism of $G$ given by conjugation by some specific element $g$ of $G$, so $x\mapsto x^{g}$. Prove that the inner automorphisms of $G$ form a normal subgroup of the group of all automorphisms of $G$.
Part III
After September 25, identify yourself in the 141100/Class Photo page! It is best (though not mandatory) if you do that on the 141100/Class Photo page itself.