(Prof.Todorov) Quiz 4 Practice Solutions Name: Dihedral Group D4

S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

Dihedral group D4 4 2 −1 1. Let D4 =< ρ, t | ρ = e, t = e, tρt = ρ > be the dihedral group.

2 3 2 3 (a) Write the Cayley table for D4. You may use the fact that {e, ρ, ρ , ρ , t, tρ, tρ , tρ } are all distinct elements of D4.

Table 1: D4 2 3 2 3 D4 e ρ ρ ρ t tρ tρ tρ e e ρ ρ2 ρ3 t tρ tρ2 tρ3 ρ ρ ρ2 ρ3 e tρ3 t tρ tρ2 ρ2 ρ2 ρ3 e ρ tρ2 tρ3 t tρ ρ3 ρ3 e ρ ρ2 tρ tρ2 tρ3 t t t tρ tρ2 tρ3 e ρ ρ2 ρ3 tρ tρ tρ2 tρ3 t ρ3 e ρ ρ2 tρ2 tρ2 tρ3 t tρ ρ2 ρ3 e ρ tρ3 tρ3 t tρ tρ2 ρ ρ2 ρ3 e

(b) Find the subgroup generated by {tρ2}. Done in class. (c) Find the subgroup generated by {t, tρ2}. First notice that the subgroup generated by {t, tρ2}, must contain t2 = e and also ttρ2 = ρ2. Next, make the Cayley table of the elements {e, t, tρ2, ρ2} and check if it is closed under operation and inverses.

2 Table 2: Subgroup < t, tρ > of D4 < t, tρ2 >) e t tρ2 ρ2 e e t tρ2 ρ2 t t e ρ2 tρ2 tρ2 tρ2 ρ2 e t ρ2 ρ2 tρ2 t e

2 2 Therefore S = {e, t, tρ , ρ } is a nonempty subset of D4 closed under operation and inverses. So S is a subgroup of D4 by one of the subgroup theorems (”two step subgroup theorem”).

(d) Find the order of each of the elements of D4. From the Table for D4 it follows that the orders of the elements of D4 are as follows (i.e. these are the smallest positive integers such that an = e): |e| = 1, |ρ| = 4, |ρ2| = 2, |ρ3| = 4, |t| = 2, |tρ| = 2, |tρ2| = 2, |tρ3| = 2.

1 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

Isomorphisms

2. Prove that the map f :(Z10, +) → (Z10, +) deﬁned by f(x) = x + 2 (mod10) is not an isomorphism of groups.

• One way: f(x) is NOT group isomorphism, since f(x + y) = 2 + x + y, but f(x) + f(y) = 2 + x + 2 + y, and 2 + x + y 6= 4 + x + y = 2 + x + 2 + y in Z10. Therefore the property f(a b) = f(a)f(b) does not hold. 0 • Another way: For each isomorphism of groups f : G → G , f(eG) = eG0 . However f(0) = 2 6= 0, hence f is not isomorphism.

3. Prove that the map f :(Z10, +) → (Z10, +) deﬁned by f(x) = 3x (mod10) is an isomorphism of groups.

• Isomorphisms between cyclic groups G =< a > and G0 =< b > of the same order can be deﬁned by – sending a, the generator of group G to a generator of G0 and – deﬁning f(ai) := (f(a))i.

• (Z10, +) is a cyclic group of order 10 with a generator 1.

• Generators of (Z10, +) are all integers {k | 1 ≤ k ≤ 9, gcd(k, 10) = 1}

• f(1) = 3 which is also a generator in (Z10, +) • f(ai) = (f(a))i in additive notation with generator 1, is written as f(i · 1) = i · f(1), which is the same as f(x) = x · 3 or f(x) = 3x. • Therefere f is an isomorphism of the above cyclic groups.

4. let G =< a > be a cyclic group of order 10. Prove that the map f : G → G deﬁned by f(a) = a3 and f(ai) = a3i is a group isomorphism.

• Isomorphisms between cyclic groups G =< a > and G0 =< b > of the same order can be deﬁned by – sending a, the generator of group G to a generator of G0 and – deﬁning f(ai) := (f(a))i. • G =< a > is a cyclic group of order 10 with a generator a. • Generators of < a > are all {at | 1 ≤ t ≤ 9, gcd(t, 10) = 1} • f(a) = a3 which is also a generator of G =< a > • From f(ai) = a3i we have f(ai) = a3i = (a3)i = (f(a))i • Therefere f is an isomorphism of the above cyclic groups.

2 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

5. Let G =< a > be a cyclic group of order 10. Prove that the map f : G → G deﬁned by f(a) = a4 and f(ai) = a4i is not group isomorphism.

• (One way) Isomorphism must send generator to a generator (see previous problems) but a4 is not generator the cyclic group of order 10, G =< a > since gcd(4, 10) = 2 6= 1. • (Another way) Orders of a and f(a) must be the same if f is an isomorphism. But 4 gcd(4,10) 2 10 |a| = 10 and |f(a)| = |a | = |a | = |a | = 2 = 5. Hence |a| 6= |f(a)|. Hence f is not an isomorphism. • (Third way) The map f is not one-to-one. Proof: f(a) = a4 and f(a7) = a14 = a4 but a 6= a7 in a cyclic group of order 10 (all elements {e, a, a2, a3, a4, a5, a6, a7, a8, a9} are distinct.

6. Let G =< a > be a cyclic group of order 10. Describe all automorphisms of G.

• Hint: f(a) must be a generator of G =< a >.

7. Describe 3 diﬀerent group isomorphisms (Z10, +) → (Z10, +). (Make sure that you explain why they are isomorphisms!)

•

8. Are the groups S3 and U(9) isomorphic? Prove your statement. • No. – Theorem: Let G ∼= G0 be isomorphic groups. Then G is abelian if and only if G0 is abelian.

– S3 is non-abelian and U(9) is abelian.

9. Are the groups S3 and D3 isomorphic? Prove your statement.

• Make a table for D3

• Make a table for S3 • Deﬁne a map: f(ρ) = (123), f(ρ2) = (132), f(ρ3) = (1), f(t) = (12), f(tρ) = (12)(123) = (23), f(tρ2) = (12)(132) = (13) • Check that the Cayley multiplication tables are the same under this identiﬁcation (func- tion).

10. Are the groups U(5) and U(10) isomorphic? Prove your statement.

• U(5) = {1, 2, 3, 4}, integers relatively prime to 5 and between 1 and 4. • 21 = 2, 22 = 4, 23 = 3, 24 = 1 implies that 2 generates all of U(5)

• U(5) is cyclic group of order 4, therefore isomorphic to (Z4, +). • U(10) = {1, 3, 7, 9}, integers relatively prime to 10 and between 1 and 9.

3 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

• 31 = 3, 32 = 9, 33 = 7, 34 = 1 implies that 3 generates all of U(10)

• U(10) is cyclic group of order 4, therefore isomorphic to (Z4, +).

• Both U(5) and U(10) are isomorphic to (Z4, +), therefore they are isomorphic.

4 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

Cosets

4 2 −1 11. Let G = D4 Let D4 =< ρ, t | ρ = e, t = e, tρt = ρ > be the dihedral group with the distinct elements: {e, ρ, ρ2, ρ3, t, tρ, tρ2, tρ3}. Let H =< t > be the subgroup generated by t.

(a) What are the elements of H? • H = {t, t2 = e} = {e, t} (b) What is the size of H? • |H| = |{e, t}| = 2 (c) What is the size of each of the left cosets of H in G. • |aH| = |H| = 2 (d) Find all left cosets of H in G. • eH = {e, t} = H • tH = {te, tt} = {t, e} = H • ρH = {ρe, ρt} = {ρ, tρ3} (use ρt = tρ3 for this and other cosets) • ρ2H = {ρ2e, ρ2t} = {ρ2, tρ2} • ρ3H = {ρ3e, ρ3t} = {ρ3, tρ} • tρH = {tρe, tρt} = {tρ, ρ3} • tρ2H = {tρ2e, tρ2t} = {tρ2, ρ2} • tρ3H = {tρ3e, tρ3t} = {tρ3, ρ} • Notice: eH = tH = H, ρH = tρ3H, ρ2H = tρ2H, ρ3H = tρH (e) How many left cosets of H in G are there? • # of (distinct) cosets of H in G = [G : H] = |G|/|H| = 8/2 = 4 • Also notice that we computed above 4 distinct cosets. (f) What is the size of each of the right cosets of H in G. • |Ha| = |H| = 2 (g) Find all right cosets of H in G. • He = {e, t} = H • Ht = {et, tt} = {t, e} = H • Hρ = {eρ, tρ} = {ρ, tρ} • Hρ2 = {eρ2, tρ2} = {ρ2, tρ2} • Hρ3 = {eρ3, tρ3} = {ρ3, tρ3} • Htρ = {etρ, ttρ} = {tρ, ρ} • Htρ2 = {etρ2, ttρ2} = {tρ2, ρ2} • Htρ3 = {etρ3, ttρ3} = {tρ3, ρ3}

5 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

• Notice: He = Ht = H, Hρ = Htρ, Hρ2 = Htρ2, Hρ3 = Htρ3 (h) How many right cosets of H in G are there? • # of (distinct) right cosets of H in G = [G : H] = |G|/|H| = 8/2 = 4 • Also notice that we computed above 4 distinct right cosets. (i) Are the left cosets of H in G the same as the right cosets of of H in G ? • Some are the same but some are not: • eH = He, tH = Ht, ρ2H = Hρ2, tρ2H = Htρ2. • ρH 6= Hρ, ρ3H 6= Hρ3, tρH 6= Htρ, tρ3H 6= Htρ3 .

6 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

12. Let G = Z12 be the group under addition (mod12). Let H =< 3 > be the subgroup generated by 3.

(a) What are the elements of H? • H = {0, 3, 6, 9} (b) What is the size of H? • |H| = |{0, 3, 6, 9}| = 4 (c) What is the size of each of the left cosets of H in G. • |a + H| = |H| = 4 (Notice additive notation!) (d) Find all left cosets of H in G. • 0 + H = {0, 3, 6, 9} • 1 + H = {1, 4, 7, 10} • 2 + H = {2, 5, 8, 11} • 3 + H = {3, 6, 9, 0} = 0 + H = 6 + H = 9 + H • 4 + H = {4, 7, 10, 1} = 1 + H = 7 + H = 10 + H • 5 + H = {5, 8, 11, 2} = 2 + H = 8 + H = 11 + H (e) How many left cosets of H in G are there? • # of (distinct) left cosets of H in G = [G : H] = |G|/|H| = 12/4 = 3 • Also notice that we computed above 3 distinct left cosets. (f) What is the size of each of the right cosets of H in G. • |H + a| = |H| = 4 (Notice additive notation!) (g) Find all right cosets of H in G. • H + 0 = {0, 3, 6, 9} = H + 3 = H + 6 = H + 9 • H + 1 = {1, 4, 7, 10} = H + 4 = H + 7 = H + 10 • H + 2 = {2, 5, 8, 11} = H + 5 = H + 8 = H + 11 (h) How many right cosets of H in G are there? • # of (distinct) right cosets of H in G = [G : H] = |G|/|H| = 12/4 = 3 • Also notice that we computed above 3 distinct right cosets. (i) Are the left cosets of H in G the same as the right cosets of of H in G ?

• YES. • 0 + H = H + 0, 1 + H = H + 1, 2 + H = H + 2, and also a + H = H + a for all a ∈ G = Z12.

7 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

13. Let G = S4 be the permutation group on four elements {1, 2, 3, 4}. Let H =< (1342) > be the subgroup generated by the permutation α = (1342).

(a) What are the elements of H? • H = {(1342), (14)(32), (1243), (1)} (b) What is the size of H? • |H| = |{(1342), (14)(32), (1243), (1)}| = 4 (c) What is the size of each of the left cosets of H in G. • |aH| = |H| = 4 (d) Find all left cosets of H in G. • (1)H = {(1342), (14)(32), (1243), (1)} = (1342)H = (14)(32)H = (1243) = H • (12)H = {(12)(1342), (12)(14)(32), (12)(1243), (12)(1)} = {(134), (1423), (243), (12)} • etc. (e) How many distinct left cosets of H in G are there? • # of (distinct) left cosets of H in G = [G : H] = |G|/|H| = 24/4 = 6 (f) What is the size of each of the right cosets of H in G. • |Ha| = |H| = 4 (g) Find all right cosets of H in G. (h) How many right cosets of H in G are there?

(i) Are the left cosets of H in G the same as the right cosets of of H in G ?

8 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

14. Let G = U(12) be the group of invertible integers (mod12). Let H =< (5) > be the subgroup generated by the element (5).

(a) What are the elements of H?

(b) What is the size of H?

(d) Find all left cosets of H in G.

(e) How many left cosets of H in G are there?

(f) What is the size of each of the right cosets of H in G.

(g) Find all right cosets of H in G.

(h) How many right cosets of H in G are there?

(i) Are the left cosets of H in G the same as the right cosets of of H in G ?

9 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

15. Let G = A4 be the group of even permutations on four elements {1, 2, 3, 4}. Let H =< (134) > be the subgroup generated by the permutation α = (134).

(a) What are the elements of H?

(b) What is the size of H?

(e) How many left cosets of H in G are there?

(f) What is the size of each of the right cosets of H in G.

(g) Find all right cosets of H in G.

(h) How many right cosets of H in G are there?

(i) Are the left cosets of H in G the same as the right cosets of of H in G ?

10 S11MTH 3175 Group Theory (Prof.Todorov) Quiz 4 Practice Solutions Name:

16. Let G be the group generated by the permutations α = (1 2 3 4) and β = (5 6 7 8 9 10) in S10. Let H =< αβ > be the subgroup of G generated by the permutation αβ = (1 2 3 4)(5 6 7 8 9 10).

(a) What are the elements of H?

(b) What is the size of H?

(d) Find all left cosets of H in G.

(e) How many left cosets of H in G are there?

(f) What is the size of each of the right cosets of H in G.

(g) Find all right cosets of H in G.

(h) How many right cosets of H in G are there?

(i) Are the left cosets of H in G the same as the right cosets of of H in G ?