4. (a) For n eZ, define multiplication mod n by ao b-a b (where indicates regular...
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).
Problem 68. Define for any 2 n є N, the set U(n)-(x| 1 x n and gcd(z, n-1} For example U(12) 1,5,7,11 Further, define n to be multiplication modulo n. For example 9 10 90 (mod 8) 2. i. Show that o is a binary operation on U/). Hint: Use the lemma from Problem 3 on your take-home exam.) ii. Pick a є N. Prove that a: 1 (mod n) has a solution (some number z є U(n)) if and...
* (9) Let n be a positive integer. Define : Z → Zn by (k) = [k]. (a) Show that is a homomorphism. (b) Find Ker(6) and Im(). yrcises (c) To what familiar group is the quotient group Z/nZ isomorphic? Explain.
5. Let Zli_ {a + bi l a,b E Z. i2--1} be the Gaussian integers. Define a function for all a bi E Zi]. We call N the norm (a) Prove that N is multiplicative. This is, prove that for all a bi, c+di E Z[i] (b) Prove that if a + r є z[i] is a unit of Zli], then Ma + bi)-1. (c) Find all of the units in Zli 5. Let Zli_ {a + bi l a,b...
1 1 point Consider the following algorithm for factoring an integer N provided as input (in binary): For i = 2 to [VN.17 i divides N, then output (i, N/). Which of the following statements is true? This algorithm is correct, but it runs in exponential time. This algorithm is not correct, because it will sometimes fail to find a factorization of Neven if N is composite This algorithm runs in sub-linear time, and always factors N it Nis composite...
Let po, P1, ...,Pn be boolean variables. Define ak = (Pk + (ak-1)), where ao = po. Prove the following boolean-algebra identity using proof by induction and the rules of boolean algebra (given below). Poan = po, for all n > 1. Equivalently this can be written out as: po · (Pn + (Pn-1 +...+(p2 + (p1 + po)...)) = po, for all n > 1. (p')=P (a) Commutative p.q=qp p+q = 9+p (b) Associative (p. 9).r=p.(q.r) (p+q) +r=p+(q +...