## Math 350: Groups, Rings and Fields Study Guide for Exam 1

The first midterm exam is on Friday, October 9, 8:45-9:50am in our regular classroom Merrill 4. (Note that it begins 15 minutes earlier than our regular class time!)

A review session, led by Zalia Rojas, will be held on Thursday, October 8, 7:00-9:00pm in Seeley Mudd 205.

The following is a chapter by chapter guide 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.

Please be sure to review homework and example problems for the chapters given below. If you would like to see additional examples and practice problems, I recommend looking at the textbook A First Course in Abstract Algebra by John B. Fraleigh, which is on reserve for this course in the Science Library.

0: While questions will not explicitly be testing the material in this section, many of these ideas are used throughout the following chapters. In particular, you should be comfortable with the basics of set notation (page 1), union, intersection, subsets, the empty (or null) set, the power set P(X) of a set X (even though technically this isnâ€™t defined until Section 2, page 20), the meaning of S ⊆ T and S = T , and how to prove such statements. You should also know standard notation for sets of numbers at the top of page 2.

1: You should know what we mean by a binary operation, and be able to determine whether it is commutative and/or associative. You should also know what we mean by the symmetric difference of two sets.

2: You must know the definition of a group! You should also be very comfortable with determining whether something is a group or not. You should know what it means for a group to be abelian, and how to prove whether a group is or isn't abelian. You should know the Division Algorithm and its proof. You should also be familiar with basic examples of groups, including sets of numbers under addition or multiplication (or even addition or multiplication modulo some number), the general linear group of invertible matrices, the trivial group and the power set under symmetric difference.

3: This section contains many great results that will make other proofs and computations much more convenient! You should know the statements and proofs of Theorems 3.1-3.6, and Ex. 3.4, which is a nice analogue of Theorem 3.5. You will not be required to use to know Theorem 3.7, but you are welcome to use it if you would like.

4: Know the definition of the order of an element and the order of a group. You should also know what we mean by a cyclic group and its generator. You should know statements and proofs of (and be able to apply) Theorems 4.1, 4.4, 4.5, 4.7 and Corollary 4.6. Know how to use the Euclidean Algorithm to obtain the gcd and the x and y mentioned in Theorem 4.2. Be familiar with the example of Klein's 4-group given at the end of this section.

5: Know the definition of a subgroup and be able to determine whether a given subset is or is not a subgroup, especially using Theorems 5.1 and 5.3. Know the statement and proof of Theorem 5.4. You should also know the results about subgroups of cyclic groups given in Theorems 5.2, 5.5, 5.7 and Corollary 5.6, and be able to use these to find subgroup lattices. In addition, you should know the definition of the center Z(G) of a group (given in Example 8 on page 46), and the centralizer Z(g) of an element of a group (given in Exercise 5.23).

6: Know the definition of a direct product and the fact that the direct product of groups gives you a group also. You should know the results given in Exercises 6.4, 6.5 and 6.6, which we discussed in class. You should also know Theorem 6.1 and be able to apply it to answer questions such as those in Exercises 6.1 and 6.2.

7: Know the definitions and notation associated with functions, including onto (surjective) and one-to-one (injective), and how to show whether a function has this attributes or not. You should also be able to tell whether a proposed function is well-defined, ie, it obeys the necessary condition of defining f(s) to be exactly one element and not more. Understand what we mean by composition and inverses, and that functions are invertible if and only if they are one-to-one and onto. Know the result stated in Theorem 7.1

8: Know the definitions and notations associated with symmetric groups, and be able to work with permutations given in cycle notation. Know the statements and proofs of Theorems 8.1, 8.2, 8.3 and Exercises 8.4, 8.8 and 8.10a. You should also know the result of Theorem 8.4, and the proof for it that we discussed in class. You should also be able to determine whether a permutation is odd or even. Finally, you should know the definition of the dihedral group and the alternating group, and be able to determine what their orders are.

This exam will be an in-class exam. It will include a mix of the following types of questions.

Conceptual/definition questions, for example:
• True or False: For a nonempty subset H of a finite group G, we only need to check if H is closed under the binary operation to determine if it is a subset.
• State what we mean by the order of an element in a group.
Proofs, especially ones that involve showing:
• a given subset is a subgroup
• a given group is abelian or cyclic
• results about orders of elements
• uniqueness
Computations/applications, such as:
• finding a subgroup lattice of a given group
• computing the product of two permutations
• computing the order of a group or element in the group