Math2070 Algebraic Structures B and C
HW2
Charles Li
Due: Feb 9th, Friday, 11:59pm
Instructions
1. Hand in your work through Gradescope before the deadline. No late HW will be accepted. You
must match the question number with page number.
2. You have to answer all the questions. Show your steps unless otherwise stated.
3. The scores for each part may be adjusted without further notification. The total score is 100.
Questions
1. Let D4 be the dihedral group consisting of the following transformations.
ˆ R0 : identity map.
ˆ R1 : rotate anticlockwise 90◦ .
ˆ R2 : rotate anticlockwise 180◦ .
ˆ R3 : rotate anticlockwise 270◦ .
ˆ FA : Reflect across line A.
ˆ FB : Reflect across line B.
ˆ FC : Reflect across line C.
ˆ FD : Reflect across line D.
Let ◦ be the composition of transformations. It is known that (D4 , ◦) is a group.
(a) Compute FA ◦ R3 . No steps are required.
(b) Compute FC ◦ FD . No steps are required.
(c) Show that (D4 , ◦) is not abelian by finding a, b ∈ D4 such that a ◦ b ̸= b ◦ a.
(d) Find the order of R1 .
(e) Find the order of FA .
2. Consider the following elements in S8 :
1 2 3 4 5 6 7
σ=
4 5 3 7 8 9 1
8
2
9
1
, τ=
6
2
2
3
3
9
4
6
5
8
6
7
7
5
8
4
9
1
For the following questions, write down your answer. Unless otherwise stated, permutations should
be written in the form of
1 2 3 4 5 6 7 8 9
.
i1 i2 i3 i4 i5 i6 i7 i8 i9
1
(a) Find σ ◦ τ and τ ◦ σ.
(b) Find σ −1 and τ −1 .
(c) Express σ and τ as products of disjoint cycles.
(d) Compute σ 2 , σ 3 . You can express your answer as products of disjoint cycles.
(e) Express σ and τ as products of transpositions.
(f) Determine if σ and τ are even or odd permutations.
(g) Find the order of σ and τ .
3. Let (G, ∗) be a group with the identity e.
(a) Let a ∈ G. Show that the ord a−1 = ord a.
(b) Let a ∈ G with order k. Suppose k is odd. Show that ord a2 is also k.
For this part only, you can use the following fact: Let k, x be positive integers. If k is odd
and k|2x, then k|x.
4. Let
σ=
1
2
2
3
3
4
4
1
∈ S5 .
(a) Express σ as a product of transpositions.
(b) Let
A = (eσ(1) |eσ(2) |eσ(3) |eσ(4) ).
Here ei ∈ R4 is a column vector with the i-th entry being 1 and other entries being 0.
Compute det(A). Show your steps.
(c) Determine whether σ is an even permutation or an odd permutation. Explain your answer
by (i) using part (a), (ii) using part(b)
5. For each of the following, determine if H is a subgroup of G.
(a) G = (Z, +), H is the set of all non-negative integers.
(b) G = (Z12 , +12 ), H = {a ∈ G : gcd(a, 12) > 1}.
(c) G = S5 , H = {e, (12), (13), (132), (123)}.
(d) G = S5 , H = {e, (12), (34), (12)(34)}.
2π
3 , s
by 2πk
3
(e) G = D5 , H = {e, r1 , s}. Here r1 is the rotation of
is the reflection about the x-axis.
(f) G = D3 , H = {e, r1 , r2 }. Here rk is the rotation
for k = 1, 2.
6. Given the following fact (you don’t need to prove the fact):
Fact: Let n be nonzero integers, m, k integers. Then n|mk if and only if
n
divisible by n if and only if k is divisible by gcd(n,m)
.
Let G = (Z15 , +Z15 ).
(a) Show that the order of m ∈ Z15 is
11.
15
gcd(15,m) .
n
gcd(n,m) |k,
i.e., mk is
Hence, or otherwise, find the order of 3, 10 and
(b) Show that m ∈ Z15 is a generator of G if and only if the order of m is 15.
(c) Find all the generators of G.
7. Suppose (G, ∗) is a cyclic group generated by g, i.e., G = ⟨g⟩. Let H be a subgroup of G.
Let
R = {r ∈ Z : g r ∈ H}.
(a) Show that 0 ∈ R.
(b) Suppose |H| > 1, show that R contains a positive integer. Let n be the smallest positive
integer in the set R.
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(c) Suppose m ∈ R. By the divisor algorithm, there exist integers q, r such that
m = qn + r, 0 ≤ r < n.
Show that r ∈ R.
(d) From the previous part, or otherwise, show that R = nZ.
(e) Suppose |H| > 1, then H = ⟨g n ⟩, where n is given by part (b).
(f) Show that H must be cyclic.
8. Let G = (Z16 , +16 ) and H = ⟨4⟩ = {0, 4, 8, 12}.
(a) Using Lagrange’s theorem, find the number of left cosets.
(b) List all the left cosets of H in G.
9. Consider the subgroup H = ⟨ (1234) ⟩ in S4 .
(a) Use the Lagrange theorem, find the number of left cosets of H in S4 .
(b) Give the list of left cosets of H in S4 .
10. Let G = S5 with composition as the binary operation. Let
H = {σ ∈ S5 |σ(1) = 1}.
(a) Show that H is a subgroup of G.
(b) Find all the left cosets of H in G.
Hint: consider (1 k) ∈ S5 .
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