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Download Definitions Abstract Algebra Well Ordering Principle. Every non
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Denitions Abstract Algebra Well Ordering Principle. Every non-empy set of Z+ contains a smallest element. Division Algorithm. If a, b ∈ Z and b > 0, (∃!q, r ∈ Z) 3 a = bq + r with 0 ≤ r < b. GCD is a linear combination. (∀a, b ∈ Z \ {0}) 3 b > 0, (∃s, t ∈ Z) 3−→ gcd(a, b) = as + bt and furthermore gcd(a, b) is the smallest integer of the form as + bt. Euclid's Lemma. If p is prime and p|ab then p|a or p|b. Fundamental Theorem of Arithmetic. (∀z ∈ Z+ ) 3 z > 1, z is prime or a unique product of primes. First PMI. Let a ∈ N. Let S ⊆ N with two properties: i) a ∈ S, ii) (∀n ∈ N)such that n ≥ a, if n ∈ S −→ n + 1 ∈ S. Then (∀n ∈ N) 3 n ≥ a, n ∈ S . Second PMI. Let a ∈ N. Let S ⊆ N with two properties: i) a ∈ S, ii) (∀n ∈ N) such that a ≤ n, ∀k 3 a ≤ k ≤ n, if k ∈ S −→ n + 1 ∈ S. Then (∀n ∈ N) 3 n ≥ a, n ∈ S . Equivalence Relations. Let a, b ∈ S and R ⊆ S × S 3 (a, b) ∈ R: 1) if (∀a ∈ S) (a, a) ∈ R reexive property 2) if (∀a, b ∈ S) (a, b) ∈ R −→ (a, b) ∈ R symmetric property 3) if (∀a, b, c ∈ S) (a, b) ∈ R ∩ (b, c) ∈ R −→ (a, c) ∈ R transitive property then R is an equivalence relation on S Equivalence class of a set S containing a: [a] = {x ∈ S | x ∼ a} Partition of a set S : a collection on non-empty disjoint sets whose union is S Equivalence classes partition set S : The equivalence classes of an equivalence relation on a set S constitute a partition of S . Conversely, for any partition P of S , there is an equivalence relation on S whose equivalence classes are the elements of P . Function (mapping) ϕ from set A (domain) to set B (range): A rule that assigns to each element a ∈ A exactly one element b ∈ B 1 2 Image of A under ϕ : A → B equals {ϕ (a) | a ∈ A} Composition of functions φϕ Given ϕ : A → B and φ : B → C , the mapping from A → C dened by φϕ (a) = φ (ϕ (a)) One-to-one ϕ : A → B having the property (∀a, b ∈ A) ϕ (a) = ϕ (b) −→ a = b Onto ϕ : A → B having the property (∀b ∈ B) (∃a ∈ A) 3 ϕ (a) = b Properties of functions α : A → B , β : B → C , γ : C → D 1) (γβ) α = γ (βα) associativity 1−1 1−1 2) if α : A −→ B , β : B −→ C , then βα is 1-1 (injective) onto onto 3) if α : A −→ B , β : B −→ C , then βα is onto (surjective) 1−1 1−1 4) if α : A −→ B , then there is a function α−1 : B −→ A, such that onto onto −1 −1 (∀a ∈ A) α α (a) = a and (∀b ∈ B) αα (b) = b Binary operation (on set G): a function that assigns to each ordered pair of elements G an element of G, i.e., f : (G × G) → G closure: the condition that members of an ordered pair from set G combine to yield a member of G group: a set G together with a binary operation with the following properties 1) associativity: (∀a, b, c ∈ G) (ab) c = a (bc) 2) uniqueness of the identity: (∀a ∈ G) (∃!e ∈ G) 3 ae = ea = a 3) uniquesness of inverses: (∀a ∈ G) (∃!b ∈ G) 3 ab = ba = e Abelian: a group having the property that (∀a, b ∈ G) ab = ba order of a group | G |, the number of elements in G order of an element | g |, the smallest n ∈ Z+ 3 g n = e subgroup H of G, H ≤ G: H⊆ G such that H itself forms a group proper subgroup H of G, denoted H < G a subgroup of H of group G that is not equal to G trivial subgroup: {e} One-step Subgroup Test If G is a group and H is a nonempty subset of G, then H ≤ G if (a, b ∈ H) −→ ab−1 ∈ H 3 Two-step Subgroup Test If G is a group and H is a nonempty subset of G, then H ≤ G if (a, b ∈ H) −→ ab ∈ H ∩ a−1 ∈ H Finite Subgroup Test If H is a nite nonempty subset of groupG, then H ≤ G if H is closed under the operation of G cyclic (∃a ∈ G) 3 G = hai = {an | n ∈ Z} hai is the cyclic subgroup of G generated by a for any element a of group G, hai = {an | n ∈ Z} ≤ G if hai = G, then a is the generator of G center of a group, denoted Z (G) Z (G) = {a ∈ G | (∀g ∈ G) ag = ga} and Z (G) ≤ G centralizer of a in G, denoted C (a) C (a) = {g ∈ G | ag = ga} and C (a) ≤ G (a ∈ G) ai = aj ←→(i) | G |= n < ∞ and n | i − j if (a ∈ G) 3 ai = aj then 1)| a |=| G |=| hai | and 2) ak = e implies | a | divides k if (a ∈ G) 3| a |= n, then ai = aj ←→(i) gcd (n, i) = gcd (n, j) if (a ∈ G) 3 G = hai ∩ | G |= n, then G = ak ←→(i)gcd (n, k) = 1 (k ∈ Zn ) 3 Zn = hki←→(i)gcd (n, k) = 1 Fundamental Theorem of Cyclic Groups 1) every subgroup of a cyclic group is cyclic 2) if | hai |= n then the order of any subgroup of hai is a divisor of n n 3) (∀k ∈ Z+ ) 3 k | n, hai has exactly one subgroup of order k, i.e., a k For subgroups of Zn only, for each positive divisor k of n, 1) the set hn/ki is a unique subgroup of order k, 2) these are the only subgroups of Zn For subgroups of Zn only, for a = k, | k |= n gcd(n,k) Green not covered in lectures. number of elements of each order in a cyclic group if d is a positive divisor of n group, the number of elements of order d in a cyclic group of order n is ϕ (d) number of elements of order d in a nite group 4 in a nite group, the number of elements of order d is divisable by ϕ (d) Euler phi function ϕ (d) 1) ϕ (1) = 1 2) (∀n > 1) ϕ (n) =| U (n) | (the number of positive integers less than n, and relatively prime to n subgroup lattice a partial ordering of all the subgroups of a group permutation of a set A a function f : A → A that is a bijection permutation group of a set A a set of permutations of A that forms a group under function composition symmetric group of order n, denoted Sn the set of all permutations of a set A, | Sn |= n! products of disjoint cycles every permutation can be written as a cycle or as a product of disjoint cycles disjoint cycles commute if two cycles α and β have no entries in common, then αβ = βα order of a cycle σ , is the length of the cycle: e.g. if σ = (153) →| σ |= 3 the order of a permutation of a nite set written in disjoint cycle form is the if | σ |= x∩ | β |= y , then | σβ |=| βσ |= lcm (x, y) e.g., | (153) (2468) |= lcm (3, 4) = 12 product of 2-cycles every permutation in Sn where n > 1 is a product of 2-cycles even (odd) permutation a permutation that can be decomposed into a product of an even (odd) number of 2-cycles the group of even permutations of Sn alternating group of degree n, denoted An order of An for n > 1, | An |= n! 2