|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
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 2-4PM, 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 for which the symmetric group contains an element of order 18?
- What is the maximal order of an element in ? (That is, of a shuffling of the red cards within a deck of cards?)
Problem 2. (Selick) Let be a subgroup of index 2 in a group . Show that is normal in .
Problem 3. Let be a permutation whose cycle decomposition consists of one 5-cycle, two 3-cycles, and one 2-cycle. What is the order of the centralizer of ?
Problem 4. (Selick) Let be a group of odd order. Show that is not conjugate to unless .
Problem 5. (Dummit and Foote) Show that if is cyclic then is Abelian.
Problem 6. (Lang) Prove that if the group of automorphisms of a group is cyclic, then is Abelian.
Problem 7. (Lang)
- Let be a group and let be a subgroup of finite index. Prove that there is a normal subgroup of , contained in , so that is also finite. (Hint: Let and find a morphism whose kernel is contained in .)
- Let be a group and and be subgroups of . Suppose and . Show that
Problem 8. (Selick) Show that any group of order 56 has a normal Sylow- subgroup, for some prime dividing 56.
Problem 9. (Qualifying exam, May 1997) Let act on by permuting the factors, and let be the semi-direct product of and .
- What is the order of ?
- How many Sylow-5 subgroups does have? Write down one of them.
Problem 10. (Selick) Show that the group of unit quaternions (, subject to and ) is not a semi-direct product of two of its proper subgroups.
Problem 11. (Qualifying exam, September 2008) Let be a finite group and be a prime. Show that if is a -subgroup of , then is congruent to mod . You may wish to study the action of on by multiplication on the left.