14:332:548, E
or 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 unde
multiplication modulo p.
2. Let G be the set of 2× 2 matrices over R of the form
(
a
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