Algebra course notes Margaret Bilu December 20, 2019 1 Contents 1 Quantifiers, sets, maps, equivalence relations 3 1.1 Sets, subsets . 3 1.2 Unions, intersections, products . 4 1.3 Maps . 5 1.4 Equivalence relations . 5 2 Integers 7 2.1 Addition and multiplication on the set of integers Z ............. 7 2.2 Divisibility . 8 2.3 Euclidean division . 9 2.4 GCD and Euclid’s algorithm . 9 2.5 Unique factorization of integers . 10 2.6 Congruence classes . 11 2.7 Units in Z=nZ .................................. 13 2.8 Back to congruences . 14 2.9 Conclusion of the chapter . 15 3 Groups 15 3.1 Laws of composition . 15 3.2 Groups . 17 3.3 Subgroups . 19 3.4 Products of groups . 20 3.5 Cyclic groups . 20 3.6 Group homomorphisms . 22 3.7 Isomorphisms . 24 3.8 Classification of groups of small order . 26 3.9 Conclusion of the chapter . 27 4 Permutation groups 28 4.1 Definition . 29 4.2 Cycles . 30 4.3 Parity of a permutation . 32 4.4 Generators of Sn and An ............................ 34 4.5 Dihedral groups . 35 4.6 Conclusion of the chapter . 36 5 Cosets and Lagrange’s theorem 37 5.1 Left and right cosets . 37 5.2 Index of a subgroup . 38 5.3 Lagrange’s theorem . 40 5.4 Some arithmetic applications of Lagrange’s theorem . 42 2 5.5 Cosets and homomorphisms . 43 5.6 Conclusion of the chapter . 45 6 Normal subgroups and quotients of groups 46 6.1 Normal subgroups . 46 6.2 Quotient groups . 47 6.3 First isomorphism theorem . 49 6.4 Correspondence theorem . 51 6.5 Conclusion of the chapter . 51 7 Further topics: towards a classification of abelian finite groups 52 7.1 Elements with prime order in an abelian group . 52 7.2 Abelian finite groups . 53 7.3 Conclusion of the chapter . 56 September 4: 1 Quantifiers, sets, maps, equivalence relations Reference: Judson Chapter 1. The quantifier 8 should be read “for all”, “for every”. The quantifier 9 should be read “there exists”. You should never use them in a sentence in English, only in mathematical sentences! 1.1 Sets, subsets A set is denoted using curly brackets f; g, • either by specifying all of its elements: A = f2; 3; 7g; B = f0; 1; 2; 3;:::g: (Here, it is implicit that the set B consists of all non-negative integers. ) • or by writing it in the form fx; x satisfies Pg where P is some property. Important examples of sets include N = f1; 2; 3;:::g positive integers, or natural numbers Z = f:::; −2; −1; 0; 1; 2;:::g integers p Q = ; p; q 2 Z; q 6= 0 rational numbers q 3 and also the set R of real numbers and the set C of complex numbers. We write a 2 A to denote that a is an element of the set A. A set A is a subset of another set B if every element of A is also an element of B. We write this A ⊂ B, or A ⊆ B. If A is not a subset of B, then we may write A 6⊂ B. We say that A is a proper subset of B if A ⊂ B but A 6= B. This is denoted A ( B, and in this case there exists an element of B which is not an element of A. To show that two sets A and B are equal, it is often most convenient to show that A ⊂ B and B ⊂ A. The empty set is denoted ?. 1.2 Unions, intersections, products For two sets A and B, we define • their union to be A [ B = fx; x 2 A or x 2 Bg . • their intersection to be A \ B = fx; x 2 A and x 2 Bg . One can also consider the union of more than two sets: for sets A1;:::;An, we denote their union and intersection by n n [ \ Ai = A1 [ ::: [ An; Ai = A1 \ ::: \ An: i=1 i=1 The first one is the set of all x belonging to at least one of the Ai. The second one is the set of all x belonging to all of the Ai. If A \ B = ?, we say that A and B are disjoint. We write A n B = fx : x 2 A and x 62 Bg: Given two sets A and B, we define the cartesian product A × B to be the set of pairs (a; b) such that a 2 A and b 2 B. A × B = f(a; b); a 2 A; b 2 Bg: More generally, we can define products of more than two sets: n Y Ai = A1 × ::: × An: i=1 n If A1 = ::: = An = A, then we write this A . 4 1.3 Maps A map f : A ! B is an operation associating to every element a 2 A a unique element of B, denoted by f(a). We denote this f : a 7! f(a). The set A is called the domain of f. The image of f, denoted by f(A), is the subset of B defined by f(A) = fb 2 B : 9a 2 A; f(a) = bg: More generally, for any subset A0 ⊂ A, we may define f(A0) = fb 2 B : 9a 2 A0; f(a) = bg: An important example of the map is the identity map id : A ! A, a 7! a, sending each element of A to itself. We may write it idA if we want to specify the set it is the identity of. The map f is said to be injective, or one-to-one, if distinct elements of A have distinct images in B. In other words: 8a; a0 2 A; a 6= a0 ) f(a) 6= f(a0); which is the same as 8 a; a0 2 A; f(a) = f(a0) ) a = a0: The map f is said to be surjective, or onto, if f(A) = B. In other words, every element of B is of the form f(a) for some a 2 A: 8b 2 B; 9a 2 A; f(a) = b: A map which is both injective and surjective is called bijective. Let f : A ! B and g : B ! C be two maps. We define the composition of f and g, denoted by g ◦ f, to be the map a 7! g(f(a)): A map f : A ! B is said to be invertible if there exists a map g : B ! A, called the inverse of f, such that g ◦ f = idA and f ◦ g = idB: A map is invertible if and only if it is bijective. 1.4 Equivalence relations A relation on a set E is a subset R ⊂ E × E. If (x; y) 2 R, we write xRy. Definition 1.4.1. The relation R is an equivalence relation if it satisfies the following: 1. (reflexive) 8x 2 E, xRx; 2. (symmetric) 8x; y 2 E, if xRy then yRx; 3. (transitive) 8x; y; z 2 E, if xRy and yRz then xRz. 5 September 9: Most of the time, we will use some special notation, like ∼ or ≡, for equivalence relations, instead of R. Example 1.4.2. We define a relation ∼ on Z given by x ∼ y if x − y is even. It is easy to check that this is an equivalence relation on Z. In fact, it is a special case of the congruence relation which we will encounter later. Definition 1.4.3. Let X be a set. A partition of X is a collection (Xi)i2I of nonempty subsets of X (indexed by some set I) which are pairwise disjoint (i.e. for all distinct i; j 2 I, Xi \ Xj = ?) and such that their union is all of X: [ Xi = X: i2I The Xi are called the parts of the partition. Example 1.4.4. Let E be the set of even integers and O the set of odd integers. Then E and O form a partition of Z. Example 1.4.5. For every letter of the alphabet, we denote by X the set of students in our class whose first name begins with . Then the collection XA;XB;:::;XZ forms a partition of the set of students of this class. Definition 1.4.6. Let ∼ be an equivalence relation on a set X. For every x 2 X, we define the equivalence class of x to be the set [x] := fy 2 X; y ∼ xg: Proposition 1.4.7. If y 2 [x], then [y] ⊂ [x]. Proof. Let z 2 [y]. Then z ∼ y. By transitivity, we have z ∼ x, so z 2 [x]. Proposition 1.4.8. Let x; x0 2 X. Then either [x] = [x0] or [x] \ [x0] = ?. Proof. If [x] \ [x0] = ? we are done. If not, there exists y 2 [x] \ [x0]. We want to show that [x] = [x0]. By definition, we have y ∼ x and y ∼ x0. By symmetry and transitivity we then have x ∼ x0, so x 2 [x0]. This implies [x] ⊂ [x0]. Similarly, we show that [x0] ⊂ [x], whence the result. Proposition 1.4.9. Let ∼ be an equivalence relation on a set X. Then the equivalence classes of ∼ form a partition of X. Conversely, every partition of the set X induces an equivalence relation on X, by definining x ∼ y if x and y lie in the same part of the partition. Definition 1.4.10. Let X be a set with an equivalence relation ∼. We define the quotient space of X for ∼ to be the set of equivalence classes of ∼, denoted by X=∼.