5520,  Fall 2004: Review Sheet for Final Exam

The test will be in roughly the same format as the midterm: there will be two parts, one on theory and one on computation.  Each part will contain 4 or 5 problems, and you will have to do two or three of them.  The exam will focus on Chapters 7-11 and 24-25; it will be comprehensive only in the sense that you must be familiar with Chapters 1-6 in order to understand the later chapters.  Go over the homework problems assigned from these chapters, and do other similar ones from the book for extra review.

The most important chapters are 7, 9, and 10: know their main theorems.  You should be comfortable with external direct products as covered in Chapter 8, but they are not as central as the material of 7, 9, and 10.  On a related note, you should know the version of Theorem 9.6 proven in class, on the "internal direct products" of two subgroups of a given group.

There will probably be one problem on Chapter 11.  You should know the main theorem of this chapter, and also how to count the number of elements of a given order in a finite abelian group.

There will probably be two problems on Chapters 24 and 25, one theoretical and one computational.  Know the Sylow theorems and how to apply them.  Theorem 24.6 on pq groups is often useful.  The final problem set gives plenty of review for this material.