For any a 2 Z∗ p = f1; : : : ; p - 1g, with p being a prime, define the function fa(x) : Z∗ p ! Z∗ p such
that fa(x) = a · x (mod p).
(a) Prove that fa(x) is a bijection.
(b) Deduce that every element of Z∗ p has a multiplicative inverse and that (Z∗ p; :) is a group under
multiplication modulo p
14:332:548, Error Control Coding Homework 2 Rutgers University 1. For any a ∈ Z∗p = {1, . . . , p − 1}, with p being a prime, define the function fa(x) : Z∗p → Z∗p such that fa(x) = a · x (mod p). (a) Prove that fa(x) is a bijection. (b) Deduce that every element of Z∗p has a multiplicative inverse and that (Z∗p, .) is a group under multiplication modulo p. 2. Let G be the set of 2× 2 matrices over R of the form ( a b 0 c ) , where a, b, c ∈ R. (a) For what condition on a, b, c is G a group under matrix multiplication? Show that G is indeed a group under this condition. (b) Let H be the set of elements in G for which a = c = 1. Prove that H is a subgroup of G. (c) Show that H is isomorphic to a group that you know. 3. Let a be an element of finite order in a group G. Let O(a) denote the order of a in G. (a) Show that for every positive integer `, a` = 1 if and only if O(a)|`. Hint: a` = ar, where r is the remainder of ` when divided by O(a). (b) Show that for every positive integer n, O(an) = O(a) gcd(O(a), n) . (c) Let a and b be elements with finite orders m and n, respectively, in a commutative group and suppose that gcd(m,n) = 1. Show that O(a · b) = mn. Hint: First verify that e = O(a · b) divides mn. Then suppose to the contrary that e < mn. ar- gue that this implies the existence of a prime divisor p of n (say) such that e|(mn/p). compute (ab)mn/p and show that it is equal both to 1 and to bmn/p. 4. for any group g, define autg to be the set of automorphisms of g. that is the set of isomorphisms from g onto itself. (a) take g = (z∗p,+) and φ(x) : g → g where φ(x) = ax + b for two given a, b ∈ g and multiplication is done modulo p. for what values of a and b, φ is an automorphism of g? (b) for any group gthe, show that autg is a group under function composition. 1 mn.="" ar-="" gue="" that="" this="" implies="" the="" existence="" of="" a="" prime="" divisor="" p="" of="" n="" (say)="" such="" that="" e|(mn/p).="" compute="" (ab)mn/p="" and="" show="" that="" it="" is="" equal="" both="" to="" 1="" and="" to="" bmn/p.="" 4.="" for="" any="" group="" g,="" define="" autg="" to="" be="" the="" set="" of="" automorphisms="" of="" g.="" that="" is="" the="" set="" of="" isomorphisms="" from="" g="" onto="" itself.="" (a)="" take="" g="(Z∗p,+)" and="" φ(x)="" :="" g="" →="" g="" where="" φ(x)="ax" +="" b="" for="" two="" given="" a,="" b="" ∈="" g="" and="" multiplication="" is="" done="" modulo="" p.="" for="" what="" values="" of="" a="" and="" b,="" φ="" is="" an="" automorphism="" of="" g?="" (b)="" for="" any="" group="" gthe,="" show="" that="" autg="" is="" a="" group="" under="" function="" composition.="">