Question

4. (a) For n eZ, define multiplication mod n by ao b-a b (where indicates regular real number multiplication), prove that On is a binary operation on Zn. That is, (Hint your proof will be very similar to the proof for homework 4 problem 7ab) (b) Let n E Z. Is the binary algebraic structure 〈L,On) always a group? Explain. (c) Prove There exists be Zn such that a n I if and only if (a, n)1. (d) It is easy to check that On is associative, and that I is the On-identity (you do not need to do this). Let aEZ (e) Let Zİ-(а є Zn I (a,n)-1). The preceeding comments and exercise shows that 〈zi, on) is a group. Give the group table for 〈ZgOJ· Is this a cyclic group? Explain. To what familiar group is Z; isomorphic?

Answer 1

The multiplication mod n is defined by a^bar circleddotoperator_n b^bar = a middot b^bar. Let, a_1^bar, a_2^bar, b_1^bar, b_2^bar elementof Z_n where, a_1^bar = a_2^bar, b_1^bar = b_2^bar now, a_1^bar circleddotoperator_n b_1^bar a_1 middot b_1 = a_1^bar middot b_1^bar = a_2^bar middot b_2^bar = a_2 middot b_2^bar = a_2^bar circleddotoperator_n b_2^bar Next we are checking the set of congruence classes n that are coprime to n satisfy the of group or not. Now, a is coprime to n if and only if gcd (a, n) = 1 Integers in the same congruence classes always satisfies that one is coprime to n it only if the other is. Therefore circleddotoperator_n is well . as circleddotoperator_n is binary operation, circleddotoperator_n is classed. As, gcd (a, n) = 1 and gcd (b, n) = 1 gcd (ab, n) = 1, hence classed. Associativity: let, a^bar, b^bar, c^bar elementof z_n (a^bar circleddotoperator_n b^bar) circleddotoperator_n c^bar = (a middot b^bar) circleddotoperator_n c^bar = a middot