## Math 251, Fall 2017, University of Vermont

Professor Sands

Homework is ordinarily due on Wednesdays at the beginning of class.
If you have questions, you can contact me using email: sands@cems.uvm.edu

Problems listed on this page are for practice and need not be handed in. The problems to be handed in are found by following the links.
Problems in parantheses are required of graduate students and optional for undergraduates, but recommended as preparation for later sections and tests.

HOMEWORK MUST BE:
1. Prefaced by a signed statement that the work on the assignment is your own.
2. Written in your own words, without any references in front of you other than the textbook and notes from class. Cite any other resources you consult, so as to give credit and allow the reader to locate the exact source.  Use of solutions written by others is not allowed, and is a violation of academic integrity.
3. Legible, complete and concise. (Typeset is ideal!)
4. Written in complete grammatical English sentences.
5. Explained fully and logically with reasoning behind each step. When using a theorem or proposition, say so explicitly.  Show all work required to arrive at your answer.
6. Self-contained: explain the use of any symbols or terms not introduced in class.  Prove any results you use that have not been established in class.

#      Due Date           Homework Assignment                            Topic
 1 Sept. 6 First  Problem Set: Follow this link.  Solutions 0.1 Basics 0.2 Properties of the Integers 2 Sept. 13 0.3 #  3, (5), 6, 7, (8), (12)     HW2 link 1.1 #   1abc, 2abc, 4, 6abc      Solutions0.3 #15a, 15b, Z/nZ, modular arithmetic  Intro. to groupsInverse of an element 3 Sept. 20 1.1 # 13, 14, 16, 19, 22, 24, 35  HW3 link  1.2 #1, (4,5), 9, (13)                   Solutions1.3 # 1, 4, 5, (10), 19, (20) Order of an element  Dihedral GroupsPermutations and Symmetric Groups 4 Sept. 27 1.4 #  2, 3, 10, (11abc)       HW4 link 1.5 #  1                                Solutions1.6 # 1, 2, 3, 8, 9, 10, 17 (19) Matrix Groups  Quaternions   Homomorphisms Test Oct. 4 Chapters 0-1     Practice             Hints 5 Oct. 11 2.1 # 3, (4), (6), 8, 10, 12   HW 5 Link  Solutions2.3 #  2, 3, 10, 11, (12), 16, (20), (21), (26) Subgroups Cyclic Groups 6 Oct. 18 2.5 #  2, 9, (12)     HW 6 Link3.2 #  2, (4), 5, 8, (11), 22 Solutions Lattice of SubgroupsCosets and Lagrange's Thm. 7 Oct. 25 3.1 #  1, 3, 4, 5, 6, 7, 11, (14), 20, 22a, (32), 42 3.3 #   2 (choose 3 parts), 3  HW 7 Link 3.4 #  1                         Solutions Quotient GroupsIsomorphism Theorems  Simple Groups Test Nov. 1 Chapters 2-3   Practice    Hints     Test Solutions 8 Nov. 8 3.5 #   2, (3), (4), 7, 9, (17)   HW 8 Link4.1 #   1, 4                             Solutions 4.2 #   3, (11) 4.3 #   2c, (5) , 8, 11b, 11c, (20) Alternating GroupGroup Actions  Cayley's Theorem Class Equation 9 Nov. 15 4.5 #  6, (18), 13, (22), (32)   HW 9 5.1 #  1 (for n=2) , 5, 14 (for n=2)5.4 #  (10), 13              Solutions Sylow Theorems Direct Products (External) Direct Products  (Internal) Break week of Nov. 22  Test Nov. 29 Chapters 4-5     Practice Problems     Hints                      Test solutions 10 Dec. 6 5.2 #  1a                 HW 10 Link 7.1 #  5abef, 6abce    Solutions 7.3 #  6, 7, 10 Finite Abelian Groups Rings  Homomorphisms and Quotients FINAL EXAM Dec. 11