Discrete Mathematics in Computer Science October 26, 2020 | B10. A Glimpse of Abstract Algebra Discrete Mathematics in Computer Science B10. A Glimpse of Abstract Algebra B10.1 Abstract Groups Malte Helmert, Gabriele R¨oger University of Basel B10.2 Symmetric Group and Permutation Groups October 26, 2020 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 1 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 2 / 23 B10. A Glimpse of Abstract Algebra Abstract Groups B10. A Glimpse of Abstract Algebra Abstract Groups Abstract Algebra Elementary algebra: \Arithmetics with variables" I p −b± b2−4ac I e. g. x = 2a describes the solutions of ax 2 + bx + c = 0 where a 6= 0. B10.1 Abstract Groups I Variables for numbers and operations such as addition, subtraction, multiplication, division . I \What you learn at school." I Abstract algebra: Generalization of elementary algebra I Arbitrary sets and operations on their elements I e. g. permutations of a given set S plus function composition I Abstract algebra studies arbitrary sets and operations based on certain properties (such as associativity). Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 3 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 4 / 23 B10. A Glimpse of Abstract Algebra Abstract Groups B10. A Glimpse of Abstract Algebra Abstract Groups Binary operations Groups Definition (Group) A group G = (S; ·) is given by a set S and a binary operation · on S that satisfy the group axioms: I Associativity:( x · y) · z = x · (y · z) for all x; y; z 2 S. I A binary operation on a set S is a function f : S × S ! S. I Identity element: There exists an e 2 S such that I e. g. add : N0 × N0 ! N0 for addition of natural numbers. for all x 2 S it holds that x · e = e · x = x. I In infix notation, we write the operator between the operands, Element e is called identity of neutral element of the group. e. g. x + y instead of add(x; y). I Inverse element: For every x 2 S there is a y 2 S such that x · y = y · x = e, where e is the identity element. A group is called abelian if · is also commutative, i. e. for all x; y 2 S it holds that x · y = y · x. Cardinality jSj is called the order of the group. Niels Henrik Abel: Norwegian mathematician (1802{1829), cf. Abel prize Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 5 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 6 / 23 B10. A Glimpse of Abstract Algebra Abstract Groups B10. A Glimpse of Abstract Algebra Abstract Groups Example: (Z; +) Uniqueness of Identity and Inverses (Z; +) is a group: Theorem Every group G = (S; ·) has only one identity element and for each I Z is closed under addition, i. e. for x; y 2 Z it holds that x 2 S the inverse of x is unique. x + y 2 Z The+ operator is associative: for all x; x; z 2 it holds that I Z Proof. (x + y) + z = x + (y + z). identity: Assume that there are two identity elements e; e0 2 S I Integer 0 is the neutral element: for all integers x it holds with e 6= e0. Then for all x 2 S it holds that x · e = e · x = x and that x + 0 = 0 + x = x. that x · e0 = e0 · x = x. Using x = e0, we get e0 · e = e0 and using I Every integer x has an inverse element in the integers, namely x = e we get e0 · e = e, so overall e0 = e. −x, because x + (−x) = (−x) + x = 0. inverse: homework assignment (Z; +) also is an abelian group because for all x; y 2 Z it holds that x + y = y + x. We often denote the identity element with 1 and the inverse of x with x−1. Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 7 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 8 / 23 B10. A Glimpse of Abstract Algebra Abstract Groups B10. A Glimpse of Abstract Algebra Abstract Groups Division { Right Quotient Division { Left Quotient Theorem Let G = (S; ·) be a group. Then for all a; b 2 S the equation x · b = a has exactly one solution x in S, namely x = a · b−1. −1 We call a · b the right-quotient of a by b and also write it as a=b. Theorem Let G = (S; ·) be a group. Then for all a; b 2 S the equation Proof. b · x = a has exactly one solution x in S, namely x = b−1 · a. −1 It is a solution: With x = a · b it holds that −1 x · b = (a · b−1) · b = a · (b−1 · b) = a · 1 = a. We call b · a the left-quotient of a by b and also write it as b n a. The solution is unique: Proof omitted Assume x and x0 are distinct solutions. Then x · b = a = x0 · b. Multiplying both sides by b−1, we get (x · b) · b−1 = (x0 · b) · b−1 and with associativity x · (b · b−1) = x0 · (b · b−1). With the axiom on inverse elements this leads to x · 1 = x0 · 1 and with the axiom on the identity element ultimately to x = x0. Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 9 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 10 / 23 B10. A Glimpse of Abstract Algebra Abstract Groups B10. A Glimpse of Abstract Algebra Abstract Groups Quotients in Abelian Groups Group Homomorphism A group homomorphism is a function that preserves group structure: Definition (Group homomorphism) Theorem Let G = (S; ·) and G 0 = (S0; ◦) be groups. If G = (S; ·) is an abelian group then it holds for all x; y 2 S A homomorphism from G to G 0 is a function f : S ! S0 such that that x=y = ynx. for all x; y 2 S it holds that f (x · y) = f (x) ◦ f (y). Proof. Consider arbitrary x; y 2 S. As · is commutative, it holds that Definition (Group Isomorphism) −1 −1 x=y = x · y = y · x = ynx. A group homomorphism that is bijective is called a group isomorophism. Groups G and H are called isomorphic if there is a group isomorphism from G to H. From a practical perspective, isomorphic groups are identical up to renaming. Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 11 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 12 / 23 B10. A Glimpse of Abstract Algebra Abstract Groups B10. A Glimpse of Abstract Algebra Abstract Groups Group Homomorphism { Example Outlook I Consider G = (Z; +) and H = (f1; −1g; ·) with I 1 · 1 = −1 · −1 = 1 I 1 · −1 = −1 · 1 = −1 ( I A subgroup of G = (S; ·) is a group H = (S0; ◦) with S0 ⊆ S 1 if x is even 0 0 I Let f : Z ! f1; −1g with f (x) = and ◦ the restriction of · to S × S . −1 if x is odd I S 0 always contains the identity element and I f is a homomorphism from G to H: is closed under group operation and inverse for all x; y 2 Z it holds that I group homomorphisms preserve many properties of subgroups ( 1 if x + y is even I Other algebraic structures, e. g. f (x + y) = I Semi-group: requires only associativity −1 if x + y is odd I Monoid: requires associativity and identity element ( 1 if x and y have the same parity I Ringoids: algebraic structures with two binary operations = I multiplication and addition −1 if x and y have different parity I multiplication distributes over addition ( e. g. ring and field 1 if f (x) = f (y) I = −1 if f (x) 6= f (y) = f (x) · f (y) Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 13 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 14 / 23 B10. A Glimpse of Abstract Algebra Symmetric Group and Permutation Groups B10. A Glimpse of Abstract Algebra Symmetric Group and Permutation Groups Reminder: Permutations B10.2 Symmetric Group and Permutation Groups Definition (Permutation) Let S be a set. A bijection π : S ! S is called a permutation of S. Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 15 / 23 Malte Helmert, Gabriele R¨oger(University of Basel)Discrete Mathematics in Computer Science October 26, 2020 16 / 23 B10. A Glimpse of Abstract Algebra Symmetric Group and Permutation Groups B10. A Glimpse of Abstract Algebra Symmetric Group and Permutation Groups Symmetric Group Symmetric Group { Proof I Theorem Theorem (Symmetric Group) For set M, Sym(M) = (fσ : M ! M j σ is bijectiveg; ·) is a group. Let M be a set. Then Sym(M)=( S; ·), where Definition (Group) I S is the set of all permutations of M, and A group G = (S; ·) is given by a set S and I · denotes function composition, a binary operation · on S that satisfy the group axioms: is a group, called the symmetric group of M.