Clear writing is essential to mathematical communication, as you probably realize from reading better and worse mathematical texts. Good exposition is an acquired and important skill. Throughout this class, you'll receive feedback on solutions to your problem sets, and you should use this to refine your ability to communicate your ideas clearly and effectively. This writing project will give you an opportunity to focus on your exposition, as opposed to absorbing new mathematical content.
Daniel Murphy (the WIM grader) will be available for help with the writing project. You can contact him at dmurphy-at-math-dot-stanford-dot-edu. His office is 381-K, on the first floor of the math building, and his office hours for WIM are simultanous with his regular office hours for 120. He will also often be available by appointment; just send him an e-mail.
The project. Your goal is to present a readable and complete discussion of the semidirect product, and how you can use it to build new groups out of old ones. Semidirect products are discussed in Section 5.5, and you should cover the material from that section that you need, rather than taking it as given. Write your own discussion of the semidirect product. In particular, be careful to avoid copying the book. Be sure to give examples, as they can make ideas much clearer!
The mathematical content of this assignment is not intended to be the primary challenge. The point of this project is to concentrate on the exposition of the topic. Your paper should be 4-7 pages long; quality, not length, is what matters. Past papers in this class have gone on to win university writing awards, and I hope to nominate at least one paper this year too.
What you should do. Your target audience is a typical Math 120 colleague who has not yet read this section. Your target audience is not me or Daniel. If you have been frustrated by reading mathematical writing in the past (which you undoubtedly have), this is your chance to show how it should be done!
In the introduction, you should describe the notion of semidirect product informally, and explain why the reader might want to know about it, and why the notion is important. Put it in some larger context. Give enlightening examples. By the end of the introduction (but not at the start!) there should be a precise definition of the semidirect product. Prepare the reader for what follows by succinctly describing the main ideas and techniques you will use. Then give a detailed dicussion. Be clear what results you are quoting, and try to use as little as possible from earlier in the text. The less self-contained the paper is, the less useful it is to the reader. Do not just say something like "by Theorem 4.2 of the book" --- state any invoked theorem precisely or else give it a descriptive name (such as "the First Isomorphism Theorem"). Your paper should be readable by someone who is familiar with the material of the text up until this section, but who learned it from a different source.
You may want a brief conclusion, in which you highlight the key points of the exposition, so your reader can remember them. This is an opportunity to make sure your reader has a big picture in mind. Ask yourself: what should the reader remember after reading this paper?
Define your notation ("let G be a group..."). You don't need to define "group", "abelian group", etc.; your target audience is familiar with these notions, and can be assumed to have read everything up until this section of the book.
Use complete sentences. Use paragraphs to organize your ideas into logical chunks. Do not use shorthand symbols and words when possible ("iff", right arrows, three dots for therefore, etc.) --- these shorthand symbols are useful for the author, and sometimes necessary during a lecture when time is in short supply, but they needlessly slow down the reader. But definitely use "usual" mathematical notation (of the sort used in the text).
Run your draft by someone else (ideally in the class).
Suggestions from Daniel. Although there is a degree of subjectivity in what constitutes good writing, the following suggestions are common practice, and while they do not necesarily need to be followed, they are provided in case some resonate with your idea of style. It would be unfortunate if points of good style were not followed only for lack of consideration. A helpful resource with several suggestions is the article with writing guidelines for MIT math majors written by Professor Kleiman, which is available here. (At the end there are several other references to guides on writing.) Here are some suggestions that Daniel has found helpful (many from the above reference):
You can use whatever word processing or typesetting program you wish. The standard one used in mathematics, statistics, and other parts of science and engineering is called LaTeX, a version of Donald Knuth's famous TeX typesetting program. Implementations of LaTeX are available for free on all operating systems. A not-so-short introduction to LaTeX is available here.
Here is a link to an article about writing mathematics well. Click on "In His Own Words".
You might want to be aware of the Hume Writing Center, which offers its services for any stage of the writing process. You can use their online schedule to make appointments of 30 or 60 minutes with one of their tutors in the main location of the Center (open M-Th 10-6 and F 10-4). You can also use their drop-in schedule to see tutors without an appointment either in the Center or at one of many remote locations, most of which operate in the evening and on the weekend; click here for more. They have a good reputation, and if you use them, I'd be interested in hearing about your experience.
Acknowledgments. Thanks to Joan Licata, Kamil Szczegot, Tracy Nance, and of course Daniel Murphy.