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Abstract Algebra Abstract Algebra Definition of fields is assumed throughout these notes. “Algebra is generous; she often gives more than is asked of her.” – D’Alembert Section 1: Definition and examples 2 Section2:Whatfollowsimmediatelyfromthedefinition 3 Section 3: Bijections 4 Section 4: Commutativity 5 Section 5: Frequent groups and groups with names 6 Section 6: Group generators 7 Section 7: Subgroups 7 Section 8: Plane groups 9 Section 9: Orders of groups and elements 11 Section 10: One-generated subgroups 12 Section 11: The Euler φ function – an aside 14 Section 12: Permutation groups 15 Section 13: Group homomorphisms 21 Section 14: Group isomorphisms 22 Section 15: Group actions 25 Section 16: Cosets and Lagrange’s Theorem 27 Section 17: RSA public key encryption scheme 29 Section 18: Stabilizers, orbits 30 Section 19: Centralizer and the class equation 32 Section 20: External direct products/sums 33 Section 21: Normal subgroups 36 Section 22: Factor (or quotient) groups 37 Section 23: The internal direct product 39 Section 24: The isomorphism theorems 40 Section 25: Fundamental Theorem of Finite Abelian Groups 41 Section 26: Sandpile groups 42 Section 27: Rings 47 Section 28: Some unsurprising definitions 48 Section 29: Something new: ideals 49 Section30:Morethat’snew:characteristicofaring 51 Section 31: Quotient (or factor) rings 51 Section 32: (Integral) Domains 52 Section 33: Prime ideals and maximal ideals 53 Section34:Divisionalgorithminpolynomialrings 55 1 Section 35: Irreducibility 57 Section 36: Unique factorization domains 60 Section37:Monomialorderings(allinexercises) 63 Section 38: Modules 68 Section 39: Finitely generated modules over principal idealdomains 70 Section 40: The Chinese Remainder Theorem 72 Section 41: Fields 74 Section 42: Splitting fields 78 Section 43: Derivatives in algebra (optional) 79 Section 44: Finite fields 80 Section 45: Appendix: Euclidean algorithm for integers 84 Section46:WorkouttheRSAalgorithmanothertime 85 3n Z Section 47: Appendix: Going overboard with factoring of X X over 3Z 86 Extras : Exams − 96 Groups 1 Definition and examples Definition 1.1 A group is a non-empty set G with an associative binary operation with the following property: ∗ (1) (Identity element) There exists an element e G such that for all a G, e a = a e = a. (Why is it called “e”? This comes∈ from German “Einheit”.)∈ (2) (Inverse∗ ∗ element) For every a G there exists b G such that a b = b a = e. We often write (G, ) to mean that G is a∈ group with operation∈ . ∗ ∗ ∗ ∗ If F is a field, such as Q, R, C, then (F, +) is a group but (F, ) is not. (Justify.) Furthermore, (F 0 , ) is a group. Also, if V is a vector space over ·F , then (V, +) is a group. (Justify.)\{ Verify} · that (Z, +) is a group, but that (N, +) is not. We will study the groups abstractly and also group the groups in some natural groups of groups (decide which of the words ”group” are technical terms). Here is a possibly new example: let G = 1, 1, i, i , and let be multiplication. Then G is a group, and we can write out its multiplication{ − − } table (Cayley∗ table): 1 -1 i -i 1 1 -1 i -i -1 -1 1 -i i i i -i -1 1 -i -i i 1 -1 2 Associativity holds because we know that multiplication of complex numbers is associative. We can clearly find the identity element and an inverse (the inverse?) of each element. Consider the set H consisting of rotations of the plane around the origin by angles 90◦, 180◦, 270◦ and 360◦. Verify that H is a group if is taken to be composition. How many elements does H have? Write its multiplication table.∗ What is its identity element? Can you find a similarity with the previous example? Exercise 1.2 Let n be a positive integer and let G be the set of all complex numbers whose nth power is 1. Prove that (G, ) is a group. What is its identity element? Can you represent this group graphically? · Exercise 1.3 Let n be a positive integer. Let G = 0,...,n 1 . For any a, b G, define a b to be the remainder of a + b after dividing by{ n. Prove− that} (G, ) is a group.∈ What is∗ its identity element? For a G, what is its inverse? This group∗ is denoted in ∈ Math 112 as Zn (read: “z n”). Later we will also see the more apt notations Z/nZ and Z nZ . (COMMENT: this is NEVER “division” by zero!) For Reed students, who are very familiar with binary properties, it seems best to first narrow down the general possibilities for groups before we look at more examples. 2 What follows immediately from the definition Theorem 2.1 Let * be an associative binary operation on a non-empty set G. Then G has at most one element e satisfying the property that for all a G, e a = a e = a. ∈ ∗ ∗ Proof. If e′ is an element of G with e′ a = a e′ = a for all a G, then ∗ ∗ ∈ e′ e = e and e′ e = e′ ∗ ∗ by the defining properties of e and e, whence e = e′. In particular, a group (G, ) has exactly one element e that acts as an identity element, and it is in fact called the identity∗ element of G. Furthermore, the inverses are also unique. Theorem 2.2 Let (G, ) be a group, a G. Then there exists a unique element b G such that b a = a b =∗e. ∈ ∈ ∗ ∗ Proof. By the inverse element axiom, such an element b exists. Let c G such that c a = a c = e. Then ∈ ∗ ∗ c = c e = c (a b)=(c a) b = e b = b, ∗ ∗ ∗ ∗ ∗ ∗ by associativity and by the property of e. 1 This unique inverse element of a is typically denoted as a− . WARNING: when the operation is +, then the inverse is written a. Beware of confusion. We also∗ introduce another bit of notation:− for a G, a0 is the identity element, if n is a positive integer, then an is the shorthand for a a∈ a, where a is written n times. n n 1 n 1 ∗ ∗···∗ Clearly if n > 0, then a = a − a = a a − . WARNING: when the operation is +, then a a a (with a being∗ written∗n times) is usually denoted as na. Beware∗ of confusion.∗ ∗···∗ 3 n 1 1 n Lemma 2.3 For any n N, (a )− =(a− ) . ∈ n 1 n Proof. By definition, (a )− is the unique element of G whose product with a in any order is e. But by associativity, n 1 n n 1 1 1 n 1 a (a− ) =(a − a) (a− (a− ) − ) ∗ n 1 ∗ ∗ 1 ∗ 1 n 1 = a − (a (a− (a− ) − )) n 1 ∗ ∗ 1 ∗ 1 n 1 = a − ((a a− ) (a− ) − )) n 1 ∗ ∗ 1 n∗ 1 = a − (e (a− ) − )) n 1 ∗ ∗1 n 1 = a − (a− ) − , ∗ which by induction on n equals e (the cases n = 0 and n = 1 are trivial). Similarly, the n 1 n 1 n product of a and (a− ) in the other order is e. This proves that (a− ) is the inverse of an, which proves the lemma. n n 1 With this, if n is a negative integer, we write a to stand for (a− )− . Theorem 2.4 (Cancellation) Let (G, ) be a group, a,b,c G such that a b = a c. Then b = c. ∗ ∈ ∗ ∗ Similarly, if b a = c a, then b = c. ∗ ∗ Proof. By the axioms and the notation, 1 1 1 1 b = e b =(a− a) b = a− (a b) = a− (a c)=(a− a) c = e c = c. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ The second part is proved similarly. 1 1 Exercise 2.5 Prove that for every a G, (a− )− = a. ∈ 1 1 1 Exercise 2.6 Let a, b G. Prove that (a b)− = b− a− . ∈ ∗ ∗ Exercise 2.7 Let G be a group, a G. Then the left translation or the left multi- ∈ plication by a is the function La : G G defined by La(x) = a x. Prove that La is a one-to-one and onto function. → ∗ Exercise 2.8 Let G be a group, a G. Then the conjugation by a is the function ∈ 1 C : G G defined by C (x) = a x a− . Prove that C is a one-to-one and onto function a → a ∗ ∗ a and that its inverse is Ca−1 . 3 Bijections We study our first family of groups. Exercise 3.1 Let X be a non-empty set and let G be the set of all one-to-one and onto functions f : X X. (You may need to review what a one-to-one and onto function is.) Then (G, ) is→ a group. Verify. What is the identity element? How do we denote the inverse of f ◦G? ∈ 4 Definition 3.2 The group as in the previous exercise is denoted SX and is called the permutation group of X. Exercise 3.3 Suppose that X has in addition some built-in topology on it (for example, as a a subset of some Rn, or with a p-adic topology, or with the discrete topology, etc). Let H be the set of all homeomorphisms, i.e., all bicontinuous one-to-one and onto functions f : X X. Then (H, ) is also a group. Verify. What is its identity element? → ◦ Exercise 3.4 Let G be the set of all linear one-to-one and onto functions f : Rn Rn. Prove that G is a group under composition. Why does it follow that the set of all invertible→ n n matrices with real entries is a group under multiplication? What is the identity element× of this group? Recall that f : Rn Rn is rigid if for all x,y Rn, f(x) f(y) = x y .
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