Math 120 Writing in the Major Paper

Clear writing is essential to mathematical communication, as you probably realise from reading better and worse mathematical texts. Good exposition is an acquired and important skill. Throughout this class, you will have received 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.

Julio Gutierrez (the WIM grader) will be available for help with the writing project. You can contact him at His office is 380-T, in the basement of the math building, and he has office hours Wed, 1-4pm.

The project. The topic is the groups PSL₂ over a finite field. Your goal is to present a readable and complete proof that these groups are simple. If you have the time, you may want to add a brief discussion on the exceptional cases when the field is very small.

This theorem is discussed in section 8.8 of Artin's Algebra. Artin's proof differs in places between the different editions of his book, so you might want to look at both editions. Here is a copy of the proof from his latest edition. Unfortunately, Artin assumes that the field does not have exactly five elements, but as PSL₂(F₅) has 60 elements, Dummit & Foote, Proposition 21 in Section 4.5 deals with the extra case. There is another approach based on the sizes of the conjugacy classes in SL₂(F) that I don't have a reference for, but you should feel free to ask me about it, or try to figure it out for yourself. For the record, the most general proof I know of (ie one that will work for PSL_n (and even more generally)) is in Bourbaki's Groupes et Algebres de Lie, Chapitres IV-VI (Ch IV, no 2.7). I don't recommend using this as a source.

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 proof. Your paper should be 4-7 pages long &endash; quality is more important than length.

What you should do. Your target audience should be thought of as either a typical Math 120 colleague who has not yet studied this material, or an equivalent student at another institution taking a similar course (with a possibly different textbook). In particular, your target audience is not me or Julio. 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, start by describing the theorem, why you think it is interesting, and what the main ideas are in the proof. You should do this with a minimal of technical jargon. Choose very carefully what to include and what not to include: if the reader stopped reading at this point, would they have understood the essential points? Conclude with a precise statement of the Theorem. 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 proof, but who learned it from a different source.

You may want a brief conclusion, in which you highlight the key points of the argument, 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?

You don't need to define "group", "normal subgroup", etc.; your target is familiar with these notions, and can be assumed to be familiar with the group theory part of the Math 120 course.

Use complete sentences. 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).

A good idea is to run your draft by someone else (ideally in the class).

Timeline