|Welcome to Math 1100!|
(additions to this web site no longer count towards good deed points)
||Notes and Links
||About This Class; Monday - Non Commutative Gaussian Elimination; Thursday - the category of groups, automorphisms and conjugations, images and kernels.
||Monday - coset spaces, isomorphism theorems; Thursday - simple groups, Jordan-Holder decomposition series.
||Monday - alternating groups, group actions, The Simplicity of the Alternating Groups, HW1, HW 1 Solutions, Class Photo; Thursday - group actions, Orbit-Stabilizer Thm, Class Equation.
||Monday - Cauchy's Thm, Sylow 1; Thursday - Sylow 2.
||Monday - Sylow 3, semi-direct products, braids; HW2; HW 2 Solutions; Thursday - braids, groups of order 12, Braids
||No class Monday (Thanksgiving); Thursday - groups of order 12 cont'd.
||Term Test; Term Test Solutions on Monday, HW3; HW 3 Solutions; Thursday - solvable groups, rings: defn's & examples.
||Monday - functors, Cayley-Hamilton Thm, ideals, iso thm 1; Thursday - iso thms 2-4, integral domains, maximal ideals, One Theorem, Three Corollaries, Five Weeks
||Monday - prime ideals, primes & irreducibles, UFD's, Euc.DomainPID, Thursday - Noetherian rings, PIDUFD, Euclidean Algorithm, modules: defn & examples, HW4, HW 4 Solutions
||Monday - R is a PID iff R has a D-H norm, R-modules, 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.
||Monday-Tuesday is UofT's Fall Break, HW5, Thursday - 1t3c5w handout cont'd, JCF Tricks & Programs handout
||Monday - JCF Tricks & Programs cont'd, tensor products, Thursday - tensor products cont'd
||End-of-Course 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
||The Final Exam
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See Non Commutative Gaussian Elimination
This assignment is due at class time on Monday, October 6, 2011.
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.
Solve the following questions.
- (Selick) If is an element of a group , the order of is the least positive number n for which (may be ). If , prove that .
- (Selick) Let be a group. Show that the function given by is a morphism of groups if and only if is Abelian.
- (Lang, pp 75) Let be a group. For , the commutator of and is . Let be the subgroup of generated by all commutators of elements of . Show that is normal in , that is Abelian, and that any morphism from into an Abelian group factors through .
- (Lang, pp 75) Let be a group. An automorphism of is an invertible group morphism . An inner automorphism is an automorphism of given by conjugation by some specific element of , so . Prove that the inner automorphisms of form a normal subgroup of the group of all automorphisms of .
After September 25, identify yourself in the 14-1100/Class Photo page! It is best (though not mandatory) if you do that on the 14-1100/Class Photo page itself.