Mathematics 1214: Introduction to Group Theory
Tutorial exercise sheet 7
1. Let n ∈ N, and let α : Z → Z be the mapping α(m) = m + 1 for m ∈ Z.
As usual, we write αk = |α ◦ ·{z
· · ◦ α} if k ∈ N, α0 = ιZ and we also define αk =
n times
(α−1 )−k if k ∈ Z with k < 0.
(a) Show that αk (m) = m + k for m, k ∈ Z.
(b) Consider the set
G = {αnd : d ∈ Z}.
[In this expression for G, the variable n is fixed, but d is a dummy variable
running over Z. Check that you understand what this means!]
Show that G is a permutation group on Z.
(c) Show that G-orbit equivalence is the same as congruence modulo n.
2. Let a, b, q, r be integers such that a = bq + r. Show that gcd(a, b) = gcd(b, r).
3. It may be shown that if a, b are any integers, then there exist integers s, t such
that as + bt = gcd(a, b).
Use this to prove that if n ∈ N and c, d ∈ Z with gcd(c, n) = 1 then there is a
solution x ∈ Z to the congruence cx ≡ d (mod n).
4. Suppose that n ∈ N and k ∈ Z with n k. Prove that for m ∈ Z we have
n m ⇐⇒ n m + k.
5. (a) Show that for any real number x and any j ∈ N, we have
xj − 1 = (x − 1)(xj−1 + xj−2 + · · · + x2 + x + 1).
(b) Deduce that if b ∈ Z with b ≥ 2 and j ∈ N, then bj ≡ 1 (mod b − 1).
(c) Show that if n ∈ N0 and a1 , a2 , . . . , an ∈ Z, then
b−1
n
X
j=0
j
aj b ⇐⇒ b − 1
n
X
aj .
j=0
(d) Is 123456789123456789123456789123456789 divisible by 9?
[Hint: take b = 10 in part (c)].