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ve203 Assignment 6

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ve203 Assignment 6

Q1. Let (G, ·) be a group and let H ≤ G. Define
X = {aH | a ∈ G}
I.e. X is the set of left cosets of H. Define ? : X × X −→ X by: for all a, b ∈ G,
(aH) ? (bH) = (a · b)H.
(i) We say that H is normal if for all h ∈ H and for all g ∈ G, ghg−1 ∈ H. Prove
that if H is normal, then (X, ?) is a group. Note that you need to check that ? is
a well-defined function.
(ii) Find an example of a group (G, ·) and H ≤ G such that (X, ?) is not a group.
(5 marks)

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ve203 Assignment 6

Q1. Let (G, ·) be a group and let H ≤ G. Define
X = {aH | a ∈ G}
I.e. X is the set of left cosets of H. Define ? : X × X −→ X by: for all a, b ∈ G,
(aH) ? (bH) = (a · b)H.
(i) We say that H is normal if for all h ∈ H and for all g ∈ G, ghg−1 ∈ H. Prove
that if H is normal, then (X, ?) is a group. Note that you need to check that ? is
a well-defined function.
(ii) Find an example of a group (G, ·) and H ≤ G such that (X, ?) is not a group.
(5 marks)
Q2. Let G be the set of 2×2 invertible real matrices and let · be matrix multiplication.
Show that (G, ·) is a group. Let
A =

0 1
−1 −1

and B =

0 −1
1 0 
Find the orders of A, B and A · B.
(4 marks)
Q3. Let G be the set of 4×4 invertible real matrices and let · be matrix multiplication.
Note that (G, ·) is a group. Find n ∈ N ∪ {∞} such that
Cn =
*


0 1 0 0
0 0 0 1
0 0 1 0
1 0 0 0


+
(2 marks)
Q4. Prove, without using the product formula for ϕ(n), that if p is prime, then
ϕ(p
k
) = p
k − p
k−1
(3 marks)
Q5 Prove that for all n ∈ N\{0}, n
3 + 2n and n
4 + 3n
2 + 1 are relatively prime.
(2 marks)
Q6. Prove that every subgroup of a cyclic group is cyclic.
(2 marks)
Q7. Show that if a, b, c ∈ N with a
2 + b
2 = c
2
, then 3 | ab.
(3 marks)
Q8. Find a generator of the group ((Z/11Z)

, ⊗11).
(2 marks)
1
Q9. Find the inverse of [12]89 in the group ((Z/89Z)

, ⊗89).
(2 marks)
Q10. What is the order of [27]56 in the group ((Z/56Z)

, ⊗56)?
(2 marks)
Q11. Draw a Cayley Table for the group ((Z/9Z)

, ⊗9). Is ((Z/9Z)

, ⊗9) cyclic?
(3 marks)
Q12. Let (G, ·) be a group and let a ∈ G be an element of order n. It follows that
haiG = Cn ≤ G. Let b ∈ haiG. Therefore b = a
s
for 0 ≤ s < n.
(i) Prove that hbiG is Cm where
m =
n
gcd(s, n)
(ii) Prove that ha
t
iG = hbiG if and only if gcd(s, n) = gcd(t, n)
(4 marks)
2