Math 5310 Modern Algebra I

Instructor: Mladen Bestvina

Office: JWB 210

Web page: http://www.math.utah.edu/~bestvina/5310

Office Hours: MWF 11-12 or by appointment

Textbook: Abstract Algebra, by I.N. Herstein, Third Edition, Wiley & Sons, Inc.

Time: MWF 12:55-13:45

Room: LCB 222

The course is an introduction to group theory. We will start by reviewing few things from set theory and basic proof techniques. All students are expected to be able to write simple proofs. In addition to the textbook I will be using other sources, e.g. handouts. If time permits, I plan to discuss some applications of group theory, for example to solving Rubik's cube.

The grades will be based on homework, two midterms and the final exam. The exam dates are:

Midterms: October 2 and November 20 (these are Fridays).

Final: Tuesday, December 15, 1-3 pm.

You can contact me by email.

Homework:

HW1, due Aug 31.

HW2 from Herstein, due Sept 9:

    p.15 #27 (with proofs, of course!),

    p.15 #28 (hint: construct a function g:S->S such that fg=1 and then use #29, which was proved in class),

    p.31 #1.

    In addition, if you got <2 (out of 3) on HW1, then do p.6 #8 as a kind of a “make up”.

HW3, due Sept 14.

HW4, due Sept 21: p.117 #2,3,6,9,10,11,12

HW5, due Sept 28: p.54 #1,2,3,11,19,20. For extra credit do #22 (you may assume that the intersection of A and B is the identity subgroup - this is always true by the theorem of Lagrange that we will cover in the next section)

HW6, due Oct 5: p.64 #9,10,12,24,31,32. For extra credit: #42.

HW7, due Oct 19: p.73 #1b,1c,6,7,14. In addition, prove that groups S₃ and D₆ are isomorphic, while S₄ and D₂₄ are not (these are symmetric and dihedral groups). For non-isomorphism you should find a property of one group not satisfied by another, e.g. abelian, the order of the group, having an element of order 3, or the number of elements of order 17.

HW8, due Oct 26: p.73 #11,12,15,17,20,27,34

HW9, due Nov 2: p.91 #2,4,6,7,8,9 Hint for #4: Construct G as a subgroup of S₇. Let a=(1234567) and b=(235)(476). Show that bab^-1=a² and that the elements aⁱb^j for i=0,1,2,3,4,5,6 and j=0,1,2 form a group.

HW10, due Nov 9: p.123 #6,8 (hint for #6: show that the product of any two 2-cycles is the product of 0,1 or 2 3-cycles).
p.83: #1,3,11. (hint for #11: choose an element in G whose image in G/N is a generator).

HW11, due Nov 16: p.96: #2,3, p.87: #2,4,7. For extra credit show that the group I⁺(D) of orientation preserving isometries of a regular dodecahedron D is isomorphic to A₅. By analogy with the cube, the following are the key steps.

• Compute the order of the group by looking at the action on the set of vertices (or edges, or faces). You will have to know the size of the orbit (in the cases above the action is transitive) and the order of a point stabilier (e.g. the stabilizer of a vertex has 3 elements, just like in the cube case; the stabilizer of a face has 5 elements).
• Construct a homomorphism I⁺(D)→ A₅ by constructing 5 objects inside the dodecahedron. For example, you can divide up the set of 20 vertices into 5 groups of 4 vertices (forming inscribed tetrahedra) and watch I⁺(D) permute these tetrahedra. Alternatively, divide up the set of 30 edges into 5 groups of 6 edges. You will have to argue that permutations of these objects induced by the group are even.
• Show that this homomorphism is injective, and therefore an isomorphism.

HW12, due Nov 23: This is a warm-up for the test. p.74: #19,21,36, p.84: #17, p.96: #1.

To paraphrase Herstein, to spare the student much further agony, no homework this week. Happy Thanksgiving!