## Math 350: Groups, Rings and Fields Study Guide for Exam 2

The second midterm exam will be given out on Monday, April 11, at the end of class. You will have until the beginning on class on Friday, April 15, to complete this take-home exam.

The following is a chapter by chapter list of topics intended to help you organize the material we have covered in class as you study for your exam. It is only intended to serve as a guideline, and may not explicitly mention everything that you need to study. While this exam will emphasize the chapters below, it is cumulative, and will therefore also cover material from earlier sections. Please be sure to review homework and example problems for the relevant chapters.

For this exam, you may use the course textbook, your class notes, your old homework sets and exam, and the solutions to these which were posted on the course Moodle page. You may not use any other outside help, including other websites or texts, and you may not discuss the exam or related mathematics with anyone else.

8: Be familiar with the definitions, notation and theorems related to the symmetric group, dihedral group, alternating group, including terms such as permutations, cycles, transpositions. Know the orders of these groups, and how to determine the order of the elements in these groups.

12: Understand what it means for a given function to be a homomorphism, isomorphism, or automorphism. You should also be able to determine whether two groups are isomorphic or not (and prove it!) by constructing an isomorphism or showing that one cannot exist. Be familiar with all of the theorems in this chapter.

9: You should be able to determine whether a relation is an equivalence relation (and in particular, whether it is reflexive, symmetric, and transitive). Given an an equivalence relation, you should be able to find its equivalence classes. You should also understand the definition and major results related to cosets, especially how they form equivalence classes under ≡H (see Theorem 9.3).

10: This section contains many great results that will make other proofs and computations much more simple! You should be familiar with all the theorems in this section up to Theorem 10.6, especially Lagrange's Theorem. You should also understand the definition of index of a subgroup. You do not need to know the material on p.94-95 relating to conjugacy classes and the class equation.

11: Know what it means for a subgroup to be normal, and be able to determine whether a given subgroup is normal, based on Theorems 11.1-11.4, and their corollaries. You should also know what a quotient group is, and how to form such a quotient group given a group and a normal subgroup. Keep in mind that functions on quotient groups need to be carefully checked to make sure they are well-defined! You do not need to know about Hamiltonian groups.

13: Know what we mean by canonical homomorphism and kernel of a homomorphism. Know the fundamental theorem of group homomorphisms (Theorem 13.2) and how to use it to show that a quotient group is isomorphic to some other group. You do not need to know the second and third isomorphism theorems.

14: Know how to find all possible distinct (ie non-isomorphic) abelian groups of finite order.

16: Know the definition of a rings and those of related terms, such as commutative rings, unity, units, zero-divisors, nilpotent, direct sum, integral domain, division ring and field. Also know the statements and proofs of the theorems and corollaries in this section.

17: Know the definitions of subring and ideal, and be able to check whether a given subset of a ring is either of these.