TRB BT Assistant - Normal Subgroups and Quotient Groups Online Test (MCQ Quiz)

1. 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.

2. 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

3. 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'

4. 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.

5. 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.

6. 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

7. 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.

8. 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.

9. 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|

10. 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.

11. 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)

12. 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).

13. 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

14. 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)

15. 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

16. 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}

17. What is the order of the quotient group ℤ₁₀ / {0, 5}?

A) 2 B) 5 C) 10 D) 4

18. 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

19. 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.

20. 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

21. 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}

22. 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.

23. 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.

24. 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.

25. 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.

26. 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

27. 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.

28. 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)

29. 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.

30. 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.

31. 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.

32. 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

33. 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.

34. 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}

35. 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

36. 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

37. 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.

38. 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

39. 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

40. 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.

41. 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.

42. 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

43. 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.

44. 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.

45. 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.

46. 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.

47. What is the order of the quotient group ℤ₁₂ / {0, 6}?

A) 2 B) 3 C) 4 D) 6

48. 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|

49. 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.

50. 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

Normal Subgroups and Quotient Groups - Online Test

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