Normal Subgroups and Quotient Groups - Question Bank

1. The set of all normal subgroups of a group G forms a:
A) Group under intersection
B) Group under union
C) Lattice under intersection and generation
D) Group under multiplication
2. Consider the group of non-zero rational numbers under multiplication (ℚ*, ×). Let N = {1, -1}. Is N a normal subgroup of ℚ*?
A) Yes, because ℚ* is abelian.
B) No, because ℚ* is not abelian.
C) Yes, because for any a ∈ ℚ*, a(1)a⁻¹ = 1 and a(-1)a⁻¹ = -1, so aNa⁻¹ = N.
D) No, because N is not closed under multiplication.
3. If G/N is isomorphic to H, and N is a normal subgroup of G, what is the relationship between the orders of these groups?
A) |G| = |N| * |H|
B) |G| = |N| + |H|
C) |N| = |G| * |H|
D) |H| = |G| / |N|
4. What is the order of the quotient group ℤ₁₂ / {0, 6}?
A) 2
B) 3
C) 4
D) 6
5. If G is a non-abelian simple group, what are its normal subgroups?
A) Only the trivial subgroup {e}.
B) Only the group G itself.
C) The trivial subgroup {e} and G itself.
D) No normal subgroups exist.
6. Let G = ℤ × ℤ and N = {(k, 0) | k ∈ ℤ}. Is N a normal subgroup of G?
A) Yes, because ℤ × ℤ is abelian.
B) No, because N is not closed under addition.
C) Yes, because N is the kernel of the projection map π₁: G → ℤ defined by π₁(x, y) = y.
D) No, because N is isomorphic to ℤ.
7. The Third Isomorphism Theorem states that if N and K are normal subgroups of G with N ⊆ K, then G/K is isomorphic to (G/N)/(K/N). This theorem is useful for understanding:
A) The structure of normal subgroups.
B) The structure of quotient groups of quotient groups.
C) The relationship between subgroups and cosets.
D) The index of subgroups.
8. If N is a normal subgroup of G, and K is a normal subgroup of N, is K necessarily a normal subgroup of G?
A) Yes, always.
B) No, for example, let G = S₄, N = A₄, K = V₄ (Klein four-group).
C) Yes, if G is abelian.
D) No, unless N = G.
9. What is the order of the quotient group D₄ / Z(D₄), where Z(D₄) is the center of D₄?
A) 2
B) 4
C) 8
D) 1
10. Let G = S₃ and N = A₃ = {e, (123), (132)}. Is N a normal subgroup of S₃?
A) Yes, because A₃ is the kernel of the sign homomorphism.
B) No, because S₃ is not abelian.
C) Yes, because A₃ has index 2 in S₃.
D) No, because (12)A₃(12)⁻¹ = {(12)e(12)⁻¹, (12)(123)(12)⁻¹, (12)(132)(12)⁻¹} = {e, (132), (123)} = A₃, but other elements might not work.
11. If G is a group and N is a normal subgroup, the elements of G/N are:
A) Elements of G.
B) Subgroups of G.
C) Cosets of N in G.
D) Homomorphisms from G to N.
12. The Second Isomorphism Theorem states that if N is a normal subgroup of G and K is a subgroup of G, then (NK)/N is isomorphic to K/(N ∩ K). This implies that if N ⊆ K, then K/N is isomorphic to:
A) K/(N ∩ K)
B) G/N
C) K/(K/N)
D) K ∩ N
13. Let G = GL₂(ℝ) and N = {A ∈ G | det(A) = 1} = SL₂(ℝ). What is the order of the quotient group G/N?
A) 1
B) 2
C) ∞
D) Cannot be determined
14. If G is a simple group, what are its only normal subgroups?
A) The trivial subgroup {e} and G itself.
B) Only the trivial subgroup {e}.
C) Only G itself.
D) All subgroups.
15. Consider the group of integers under addition (ℤ, +). The subgroup 5ℤ is normal. What is the quotient group ℤ/5ℤ isomorphic to?
A) ℤ
B) ℤ₅
C) ℤ₁₀
D) ℤ/ℤ
16. Which theorem directly relates normal subgroups to group isomorphisms?
A) Lagrange's Theorem
B) Cayley's Theorem
C) The First Isomorphism Theorem
D) Sylow's Theorems
17. If N is a normal subgroup of G, and G/N is isomorphic to the trivial group {e}, what can be concluded about N?
A) N = {e}
B) N = G
C) N is a subgroup of G
D) N is G and {e}
18. Let G = ℤ/nℤ. Which subgroup is normal?
A) Any subgroup of order k where k divides n.
B) The subgroup generated by the identity element.
C) The subgroup generated by n-1.
D) All subgroups are normal because ℤ/nℤ is abelian.
19. Is the commutator subgroup G' always a normal subgroup of G?
A) Yes
B) No
C) Only if G is abelian
D) Only if G is simple
20. What is the commutator subgroup G' of a group G?
A) The smallest normal subgroup of G.
B) The subgroup generated by all commutators xyx⁻¹y⁻¹ for x, y ∈ G.
C) The center of G.
D) The kernel of every homomorphism from G.
21. If G/N is a cyclic group, does it imply that G is cyclic?
A) Yes, always.
B) No, for example, G = S₃ and N = A₃.
C) No, for example, G = D₄ and N = Z(D₄).
D) Yes, if N is also cyclic.
22. Let G = D₃ (dihedral group of order 6) and N = {e, r, r²} (rotations). Is N a normal subgroup of D₃?
A) Yes, because N is the center of D₃.
B) No, because D₃ is not abelian.
C) Yes, because N is a subgroup of index 2.
D) No, because sr s⁻¹ = sr s ≠ N.
23. Consider a group homomorphism φ: G → H. The image of φ, Im(φ), is always isomorphic to which quotient group?
A) G / Ker(φ)
B) G / Im(φ)
C) H / Ker(φ)
D) H / Im(φ)
24. If G is a group and N is a normal subgroup, the order of an element gN in the quotient group G/N is:
A) The order of g in G.
B) The order of N in G.
C) The smallest positive integer k such that (gN)ᵏ = N.
D) The order of G divided by the order of N.
25. Let G = ℤ₁₅ and N = {0, 3, 6, 9, 12}. What is the order of the quotient group G/N?
A) 3
B) 5
C) 15
D) 1
26. Which of the following is a consequence of N being a normal subgroup of G?
A) Every element in N commutes with every element in G.
B) Every element in G commutes with every element in N.
C) Every left coset is equal to its corresponding right coset.
D) G/N is isomorphic to G.
27. Let G = S₄ and N = A₄. Is N a normal subgroup of G?
A) Yes, because A₄ is the kernel of the determinant homomorphism for GL₂(ℝ).
B) No, because S₄ is not abelian.
C) Yes, because A₄ has index 2 in S₄, and subgroups of index 2 are always normal.
D) No, because S₄ is not a normal subgroup of A₄.
28. The set of all cosets of a normal subgroup N in G forms a group under the operation (aN)(bN) = (ab)N. This operation is well-defined if and only if:
A) G is abelian.
B) N is the trivial subgroup.
C) N is a normal subgroup.
D) N has finite index in G.
29. Let G = A₄ (the alternating group on 4 elements) and N = {e, (12)(34), (13)(24), (14)(23)}. Is N a normal subgroup of A₄?
A) Yes, because N is the Klein four-group and A₄ is not abelian.
B) No, because N is not closed under A₄ multiplication.
C) Yes, because N is the center of A₄.
D) No, because N has order 4 and A₄ has order 12.
30. What is the kernel of the natural projection homomorphism π: G → G/N, where N is a normal subgroup of G?
A) G
B) N
C) G/N
D) {e}
31. If N is a normal subgroup of G, and K is a subgroup of G such that N ⊆ K, when is K/N a normal subgroup of G/N?
A) Always
B) Never
C) Only if K is normal in G
D) Only if G is abelian
32. Consider the group of quaternions Q₈ = {±1, ±i, ±j, ±k}. Is the subgroup H = {±1} a normal subgroup of Q₈?
A) Yes, because H is the center of Q₈.
B) No, because Q₈ is not abelian.
C) Yes, because H is a subgroup of order 2.
D) No, because iHi⁻¹ = {i(1)i⁻¹, i(-1)i⁻¹} = {1, -1} = H, but jHj⁻¹ = {j(1)j⁻¹, j(-1)j⁻¹} = {1, -1} = H, and kHk⁻¹ = {k(1)k⁻¹, k(-1)k⁻¹} = {1, -1} = H, so it IS normal.
33. The center of a group G, Z(G) = {z ∈ G | zg = gz for all g ∈ G}, is always a normal subgroup of G. True or False?
A) True
B) False
C) Only if G is abelian
D) Only if G is finite
34. What is the order of the quotient group ℤ₁₀ / {0, 5}?
A) 2
B) 5
C) 10
D) 4
35. Let G = ℤ₆ and N = {0, 3}. Which of the following is a coset of N in G?
A) {0, 1}
B) {0, 3}
C) {1, 4}
D) {0, 2}
36. Which condition is NOT equivalent to N being a normal subgroup of G?
A) gNg⁻¹ = N for all g ∈ G
B) gN = Ng for all g ∈ G
C) gNg⁻¹ ⊆ N for all g ∈ G
D) The set of left cosets G/N forms a group
37. If N is a normal subgroup of G, then for any two elements aN and bN in G/N, their product is defined as (aN)(bN) = ?
A) aN * bN
B) (ab)N
C) abN
D) N(ab)
38. What is the identity element in the quotient group G/N?
A) The identity element of G, e
B) The identity coset, eN = N
C) Any element of N
D) The inverse of N
39. Let G = GL₂(ℝ) (the general linear group of 2x2 invertible matrices with real entries) and N = SL₂(ℝ) (the special linear group). Is N a normal subgroup of G?
A) Yes, because det(g) = 1 for all g in N.
B) No, because matrix multiplication is not commutative.
C) Yes, because for any A in G and B in N, det(ABA⁻¹) = det(A)det(B)det(A⁻¹) = 1 * det(B) * 1 = det(B) = 1, so ABA⁻¹ is in N.
D) No, because SL₂(ℝ) has index 2 in GL₂(ℝ).
40. The First Isomorphism Theorem for groups states that if φ: G → H is a surjective group homomorphism, then G/Ker(φ) is isomorphic to which group?
A) Ker(φ)
B) G
C) H
D) The image of φ, Im(φ)
41. Let G = D₄ (the dihedral group of order 8) and N = {e, r², s, sr²} where r is rotation and s is reflection. Is N a normal subgroup of D₄?
A) Yes, because N is the center of D₄.
B) No, because N is not closed under the group operation.
C) Yes, because N is a subgroup and D₄ is abelian.
D) No, because D₄ is not abelian.
42. What is the order of the quotient group G/N if G is a finite group and N is a normal subgroup of G?
A) |G|
B) |N|
C) |G| / |N|
D) |G| * |N|
43. Let G = ℤ₁₂ (the cyclic group of order 12) and N = {0, 4, 8}. Is N a normal subgroup of ℤ₁₂?
A) Yes, because ℤ₁₂ is abelian, so all its subgroups are normal.
B) No, because the order of N (3) does not divide the order of ℤ₁₂ (12).
C) Yes, because 4 + k + 8 is always congruent to k + 4 + 8 mod 12.
D) No, because 1 + N ≠ N + 1.
44. If G is an abelian group, what can be said about all of its subgroups?
A) Only the trivial subgroup is normal.
B) Only G itself is normal.
C) Every subgroup is normal.
D) No subgroup is normal unless it's the trivial subgroup or G.
45. Consider the group of integers under addition, (ℤ, +). Which subgroup is normal?
A) The subgroup generated by 2, denoted by 2ℤ
B) The subgroup generated by 3, denoted by 3ℤ
C) Any subgroup of the form nℤ for some integer n
D) All of the above
46. If φ: G → H is a group homomorphism, what is the relationship between the kernel of φ, Ker(φ), and normal subgroups?
A) Ker(φ) is always a subgroup of G, but not necessarily normal.
B) Ker(φ) is always a normal subgroup of G.
C) Ker(φ) is a normal subgroup of H.
D) Ker(φ) is isomorphic to H.
47. Let G = S₃ (the symmetric group on 3 elements) and N = {e, (12)}. Is N a normal subgroup of S₃?
A) Yes, because N is a subgroup and (12)N = N(12).
B) No, because (13)N(13)⁻¹ = {(13)e(13)⁻¹, (13)(12)(13)⁻¹} = {e, (23)} ≠ N.
C) Yes, because N is a subgroup and all elements of N commute with all elements of S₃.
D) No, because N is not a subgroup of S₃.
48. Which of the following is NOT always a normal subgroup of a group G?
A) The trivial subgroup {e}
B) The group G itself
C) The center of G, Z(G)
D) The commutator subgroup G'
49. If N is a normal subgroup of G, then the set of left cosets G/N forms a group under the operation (aN)(bN) = (ab)N. What is this group called?
A) Direct Product Group
B) Quotient Group
C) Factor Group
D) Both Quotient Group and Factor Group
50. What is the fundamental property that defines a normal subgroup N of a group G?
A) For every element g in G, gN = Ng.
B) N is a subgroup of G and for every g in G, gNg⁻¹ is a subset of N.
C) N is a subgroup of G and for every g in G, gNg⁻¹ = N.
D) N is a subgroup of G and for every n in N, gn = ng for all g in G.
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