Modern Algebra 1

Fall 2014

sage: AlternatingGroup(4).cayley_graph().show3d()

**Official description:**Studies group theory and ring theory. Prereq., MATH 3140. Undergraduates need instructor consent. Prerequisites: Restricted to graduate students only.**My unofficial description:**This course sequence is designed to serve our entering first year graduate students. It covers the basics of abstract algebra from a viewpoint appropriate to students who will be going on to research. The prerequisite is a certain level of that elusive but important thing called `mathematical maturity,' which comes from experience (e.g. proofwriting). But it does not require any specific prior knowledge beyond standard linear algebra. An important goal is to prepare for the algebra prelim exam. The treatment will be standard: we will cover approximately chapters 1-9 in Dummit and Foote.**Groups and Rings:**In mathematics it is often more useful to study the maps between mathematical objects than the objects themselves. Indeed, we will study such maps between groups (and between rings) extensively. But even more to the point, both groups and rings actually arise most naturally as a collection of self-maps of some object. For example, the symmetries of a cube (the ways you can rotate or reflect it to obtain an identical cube) form a group. We will emphasize the study of such group "actions" on sets. As for rings? The collection of endomorphisms of an abelian group is naturally a ring. (Axiomatizing the definitions of groups and rings is something that comes later, at least philosophically, and serves to organize the theory.)

- Grading breakdown: Homework 50%, Midterm 20%, Final 30%.
- Textbook: Dummit and Foote, Abstract Algebra, 3rd edition.
- Other helpful books: Hungerford, Algebra. Lang, Algebra. Jacobson, Basic Algebra I.

**Collaboration:**I encourage collaboration on homework, with three caveats. First, always try the problems by yourself first. Second, write up individual solutions after the fact,**without using crib notes**of those discussions (besides your newly improved neuronal pathways). Third, please note who you worked with on your homework.**Other resources:**If you are still stuck (but only after a good faith effort), you can consult resources (many exercises have solutions online, or in the library, available to the clever searcher). For each problem on which you were required to consult resources, you should cite which resources were used and how, very specifically (e.g. page number and book, used for one direction of proof only, student who explained solution, etc.). It's also possible you have seen some of these results in a previous course or book; if so, please specify. Again, you may consult these resources in order to learn but then must write up solutions from your understanding**without using notes**. If you do these things you will receive full credit.**LaTeX:**You are**required**to LaTeX your assignments. Getting fast at LaTeX'ing is a skill you will need to cultivate; start now. There are LaTeX resources listed below.**Grading:**Some homework will be graded by your peers, and some by myself. The aim is to*write well*, not just correctly. We will talk more about this in class.**Homework Due Monday Sept 8th:**1.1: 5, 7, 9, 20, 21, 22, 24, 25, 30 (see Example 6 p.18), 32; 1.3: 5, 6, 11, 15; 2.1: 1(d), 2(b), 5. No penalty yet for not LaTeX'ing this one, but get started!**Homework Due Monday Sept 15th:**1.2: 3, 7; 1.6: 4, 19; 1.7: 8, 10(a), 14, 18, 19. Still not penalty for not LaTeX'ing, but you should be practicing.**Homework Due Monday Sept 22nd:**- 2.1: 8; 2.2: 6 (note in part (a) they mean "containment is not necessarily true"), 10, 14;
- 2.3: 25;
- 2.4: 12 (please try to give me a good proof; for heavens sake, don't just write out the multiplication tables, that's tedious!), 15, 16.
- Problem A: Draw (and prove that it is correct), the full diagram of subgroups of D_8 (the answer is in Dummit and Foote, please do it yourself before looking it up).
- Note: LaTeX this assignment. You can do the diagram by hand, or may I suggest you use xymatrix (part of xypic), which is a very easy way to make such diagrams.

**Homework Due Monday Oct 1st (NOTE THE DUE DATE HAS BEEN DELAYED):**- Problem B (another cool way to think of when a subgroup is normal): Let H be a subgroup of G. Consider the subset R of GxG consisting of pairs that are representatives of the same left coset of H (this is an equivalence relation). Show that R is a subgroup of GxG if and only if H is a normal subgroup in G.
- 2.5: 14 (just diagram, no justification);
- 3.1: 9, 14, 19, 22, 33, 36;
- 3.2: 4, 11, 14, 18, 22.
- This one is a bit long, sorry, there's just so much good stuff to do before the midterm! Final note: please, on this assignment, don't put your name. Instead, put a four digit number of your choosing (and email me what it is with subject "My 6130 Number"). We will trade assignments to do a peer-grading exercise.

**Homework Due Wednesday Oct 8th (NOTE DELAYED DATE):**- 3.3: 6, 7, 10.
- 3.4: 2, 5.
- 3.5: 7 (rigid motions => no reflections), 12.
- 4.1: 9.
- 4.2: 8.
- 4.3: 13.

**Midterm Monday Oct 13th 6-8 pm:**(Location: ECCR 135) Three prelim problems in 2 hours. Practice prelims are here.**Homework due date to be discussed:**Do the midterm problems out on your own time nicely in LaTeX. Important: This is a non-collaborative homework. You may use Dummit and Foote but NO other resources (basically a take-home open-D&F version of the same midterm).**Homework Due Monday Oct 20th:**None, you deserve a break.**Homework Due Monday Oct 27th:**- 5.2: 6 (look back in the section for the definition of 'exponent'), 8, 9.
- 5.5: 1, 2, 6, 10.

**Homework Due Wednesday Nov 5th (NOTE DELAY):**- 4.4: 6, 8.
- 4.5: 30, 32, 44.
- 4.6: 4.
- 5.2: 1.
- 5.4: 19.

**Homework Due Monday Nov 10th:**5.5: 12, 23; 6.1: 10. And do this extra problem.**Homework Due Monday Nov 17th:**- 7.1: 6 (just say yes/no and if no, counterexample to one of the properties), 11, 14, 23, 24
- 7.2: 2, 5
- 7.3: 8, 9, 10 (for 8--10, just say yes/no and if no, counterexample to one of the properties), 14, 22, 24

**Homework Due Monday Dec 1st:**- 7.4: 7, 8, 10, 11, 13, 15, 35
- 7.5: 5
- 7.6: 1

**Homework Due Monday Dec 8th:**- 8.1: 3 (they mean the Euclidean norm, the one that makes the Euclidean property true), 7, 11.
- 8.2: 1, 6, 7 (the comment "cf" in part (b) is just informational, not a hint), 8.
- 8.3: 1, 2, 6, 8, 11.

**Homework Due Friday Dec 12th (NOTE DAY):**None.**Homework That is not due (but you should do):**7.4: 36; 9.1: 4, 6, 10, 13; 9.2: 1,2,3,4, 7,10; 9.3: 3, 9.4: 1,2,7,12.**Final Exam Thur Dec 18th, 10:30 am - 1 pm:**Kobel 330 (our usual room)

**My Advice on Studying in Graduate School****Grading Rubric****Sage**- Free, sophisticated math software for indepth exploration.- CU Sage Server - you may sign up for an account from university IP addresses.
- Ask Sage - question and answer service from the hive mind
- Sage's Quick Reference Sheets to keep by your side.

**Campus Computer Labs**- Use Sage (via link above), Latex, Maple, Mathematica, Matlab etc.- Search by software - e.g. tex, mathematica

**Latex**- the only reasonable way to type math.- Introduction to Latex - installing and how to type.
- Detexify - To find a symbol you want to know.
- WriteLatex.Com - No need to install software or visit the lab - typeset online!
- Templates: tex file and expected output.
- Tex Stackexchange - question and answer service from the hive mind