Goals & Structure of Course

Transcription

1 STUDY GUIDE: MAT 370 PROFESSOR: DAMIEN PITMAN TEXT: A FIRST COURSE IN ABSTRACT ALGEBRA BY: FRALEIGH Goals & Structure of Course Content Goals. This course serves as an introduction to modern mathematics, where by modern I mean since the 18th century. Imagine that you have lived your whole life inside a cabin. You have never been outside and the few stories you have heard of the outside world are too vague to make any sense of. A first course in abstract algebra is sort of like peeking out through the window of that cabin where everything you know of mathematics is inside the cabin and modern mathematics is on the outside. Your goal is to gain some idea of what s out there so that you have a better perspective on, and understanding of, what lies inside the cabin. You may even find the outside world to be beautiful. I think it s fair to say that studying abstract algebra for the first time is one of my favorite academic experiences; and it is partly responsible for convincing me to get my doctorate in mathematics. By learning abstract algebra you get to see how mathematical structures can be constructed from some basic principles. All you need to do your own constructions is an understanding of these basic principles. We will start by reviewing set theory. Then we will explore the notion of performing operations on the elements of the set. We will then examine the structure that arises from the definitions of these operations. General Goals. This course should reinforce much of the mathematics that you have learned thus far in your life. This means that you are expected to use what you have previously learned. This course should also develop your problem solving and proof writing skills. This means that I will go out of my way to challenge you to consider nontrivial questions. In my opinion, a good college course should make you feel like you took the bull by the horns and wrestled your way to a greater understanding of life. Personal Goals. Make up your own. 1

2 Course Structure : Reading the textbook is part of your homework and this document is your guide for that reading as well as the homework problems that you are to turn in. : A typical class day will consist of two or three distinct segments. For the first several minutes of class, we will have a sort of study hall, during which time you should ask questions individually and work on homework from sections that have already been discussed in class. After this, we will have a discussion and work examples from the section that was assigned reading for the day. Lastly, there could be a five minute quiz that will test your knowledge or understanding of a vocabulary word. The word could be any vocabulary word from the assigned reading for that class day or any previous class day. The discussion and examples will rely on a basic understanding of the assigned reading, which is why I reserve the option to quiz you over the vocabulary at any time. : I recommend making a two-column reference sheet with every vocabulary words in one column and the definition in the other. : Do your best to complete all homework. Keep in mind that the goals for homework are 1) to develop a practical understanding of the concepts from the reading, and 2) to learn and practice the technical skills that will enable you to apply your understanding of these concepts. : While reading or working problems, write out any questions you have for me. : Homework content is given in this document, but the timeline will be given in class. 2

3 Notation := means defines, e.g., R 2 := {(x, y) x, y R X + := {x X x > 0} X := {x X x = 0}! means there exists a unique element a is the inverse of a (textbook) a 1 is the inverse of a (multiplication) a is the inverse of a (addition) GL(n, R) is the general linear group of n n real, invertible matrices 0. Sets and Relations Vocabulary. set, element, empty set ( ), well defined set, subset, proper vs. improper subset (and vs. ), cartesian product, relation, function, map, mapping, domain, codomain, range, cardinality, one-to-one correspondence (bijection), one-toone, onto, composition, inverse function, same cardinality, disjoint, partition, cells, residue classed modulo n, equivalence relation, congruence modulo n Results. functions: invertible bijective Homework Problems. 6, 8, 12(d,e,f), 14, 24, 26, 30, 34 Part I: Groups ans Subgroups 1. Introduction and Examples Vocabulary. complex numbers C, imaginary numbers, absolute value, addition modulo 2π, n th roots of unity Results. Eulers s Formula Homework Problems. 2, 4, 6, 10, 16, 18, 32, 34, Binary Operations Vocabulary. binary operation, closed (under ), induced operation, commutative, associative, binary operation table, not well defined Results. Associativity of Composition, words of warning (pg. 24) Homework Problems. 6, 8, Isomorphic Binary Structures Vocabulary. binary algebraic structure, isomorphism, homomorphism property, structural property, identity element, Results. How to show that binary structures are isomorphic, Uniqueness of Identity Element, Isomorphism Property of the Identity Element, Homework Problems. 2, 6, 12, 16 3

4 4. Groups Vocabulary. group, inverse, abelian, general linear group of degree n (GL(n, R)), group tables Results. Vector space with vector addition is a group, matrices with addition is a group, matrices with multiplication is not a group, left and right cancellation laws (not necessarily true outside groups), existence and uniqueness of solution to a x = b and y a = b (x = y is possible!), existence and uniqueness of inverses, Homework Problems. 2, 6, 8, 10, 14, 20, Subgroups Vocabulary. addition and multiplication (pg. 49), order of a group, subgroup, proper vs. improper subgroups, trivial vs. nontrivial subgroups, Klein 4-group, cyclic (sub)group, generator, Results. a subset is a subgroup iff..., cyclic subgroups are smallest (and minimal), Homework Problems. 4, 6, 8, 22, 54 Group Explorer Assignment #1. See course webpage. 6. Cyclic Groups Vocabulary. order of an element, quotient, remainder, gcd, relatively prime, Results. cyclic groups are abelian, division algorithm, subgroups of cyclic groups are cyclic, subgroups of Z, relatively prime division, characterization of cyclic groups, characterization of subgroups of finite cyclic groups Homework Problems. 2, 4 (also write the gcd of n and m as a Z-linear combination like we did in class), 18, 20, 22, 24, 38, 48* * Hint for 48: Try proving the contrapositive statement: If G is infinite, then G has infinitely many subgroups. It would be enough to show that G has infinitely many distinct cyclic subgroups to do this. Think about the integers as one example of what could happen, but this is not the only possible case. Group Explorer Assignment #2. See course webpage. 7. Generating Sets and Cayley Digraphs Vocabulary. intersection of sets, generates, generators, finitely generated, vertex, arc, digraph, Cayley Digraph Results. intersection of subgroups is a subgroup, elements of the group generated by a finite set Homework Problems. 2, 4, 8, 10 4

5 Part II: Permutations, Cosets, and Direct Products 8. Groups of Permutations Vocabulary. permutation, symmetric group (S n ), dihedral group (D n ), image of H under f ( f (H)), Results. set of all permutations of a set A (S A ) is a group, Cayley s Theorem and Lemma, Homework Problems. 2, 4, 6, 8, Orbits, Cycles, and the Alternating Groups Vocabulary. orbits of σ, cycle, length of a cycle, transposition, even permutation, odd permutation, alternating group A n Results. every permutation of a finite set is a product of disjoint cycles, every permutation of at least two elements is a product of transpositions, no permutation is both even and odd, Homework Problems. 8, 10, Cosets and the Theorem of Lagrange Vocabulary. left coset, right coset, index (G : H) of H in G Results. Theorem of Lagrange, every group of prime order is cyclic, the order of an element of a finite group divides the order of the group, (G : K) = (G : H)(H : K) Homework Problems. 2, 4, 6, 8, 14, 28, 30, 32 Group Explorer Assignment #3. See course webpage. 11. Direct Products and Finitely Generated Abelian Groups Vocabulary. direct product, direct sum, least common multiple (lcm ), decomposable vs. indecomposable Results. Z m Z n = Zmn iff..., general version of last, order of elements in a direct product, FTAG!, square free order implies cyclic Homework Problems. 4, 8, 12, 16, 22, 46 5

6 Part III: Homomorphisms and Factor Groups 13. Homomorphisms Vocabulary. homomorphism, trivial homomorphism, linear transformation, projection map, reduction modulo n, image of set under a homomorphism (ϕ(a)), range of a homomorphism, inverse image, kernel, normal subgroup, Results. identity, inverses, and subgroups under homomorphisms, Ker(ϕ) = {e} iff..., homomorphism is an isomorphism iff..., kernels are normal subgroups Homework Problems. 2, 4, 6, 16, 18, 48 Group Explorer Assignment #4. See course webpage. 14. Factor Groups Vocabulary. factor group (both in terms of homomorphisms and normal subgroups), quotient group, Results. isomorphism of factor group with image of the homomorphism, cosets of normal subgroups form a group (called the factor or quotient group), fundamental homomorphism theorem (isomorphism theorem), normal subgroup conditions, Homework Problems. 2, 4, 6, 8, 10 Group Explorer Assignment #5. See course webpage. Part IV: Rings and Fields 18. Rings and Fields Vocabulary. ring, left and right distributive laws, ring homomorphism and isomorphism, commutative ring, unity, multiplicative inverse, unit, division ring, field, subring, subfield, Homework Problems. 2, 4, 8, 10, 12, 20, Integral Domains Vocabulary. 0 divisor (divisor of 0), cancellation laws, integral domain, characteristic Results. 0 divisors in Z n, multiplicative cancellation holds iff..., Venn diagram of types of rings, Homework Problems. 2, 4, 6, 10, 14 6