TRB PG Assistant - Definition of groups - subgroups, homomorphism and isomorphism, group representations, irreducible representations, unitary representations Online Test (MCQ Quiz)

1. What is the fundamental property that defines a set G as a group under a binary operation *?

A) The operation * must be commutative. B) The operation * must be associative. C) There must exist an identity element and every element must have an inverse. D) The set G must be finite.

2. Which property states that for any elements a, b, and c in a group G, (a * b) * c = a * (b * c)?

A) Closure property B) Associative property C) Identity element property D) Inverse element property

3. In a group (G, *), what is the identity element 'e' characterized by?

A) e * a = a for all a in G B) e * a = e for all a in G C) a * a = e for all a in G D) e * e = a for some a in G

4. For every element 'a' in a group G, there exists an element 'a⁻¹' such that a * a⁻¹ = a⁻¹ * a = e, where 'e' is the identity element. This is known as the:

A) Closure property B) Associative property C) Inverse property D) Commutative property

5. What is the closure property in the context of a group (G, *)?

A) For any a, b in G, a * b = b * a. B) For any a, b in G, a * b is also in G. C) For any a in G, there exists a⁻¹ in G such that a * a⁻¹ = e. D) There exists an element e in G such that a * e = a for all a in G.

6. A set G with a binary operation * forms a group if it satisfies closure, associativity, existence of an identity element, and existence of inverse elements. What is this set of axioms called?

A) Rings Axioms B) Fields Axioms C) Group Axioms D) Vector Space Axioms

7. If for every pair of elements a, b in G, a * b = b * a, the group is called:

A) Non-abelian B) Abelian or Commutative C) Cyclic D) Finite

8. Consider the set of integers Z under addition. Is this set a group?

A) No, because it lacks closure. B) No, because it lacks an identity element. C) Yes, it satisfies all group axioms. D) No, because it lacks inverse elements.

9. What is a subgroup?

A) Any subset of a group. B) A subset of a group that is itself a group under the same operation. C) A superset of a group. D) A group with fewer elements than the original group.

10. Let H be a subset of a group G. Under what condition is H a subgroup of G?

A) H is non-empty and for all a, b in H, a * b⁻¹ is in H. B) H is non-empty and for all a, b in H, a * b is in H. C) H is non-empty and contains the identity element of G. D) H is non-empty and all elements of H have inverses within H.

11. The trivial subgroup of any group G is:

A) The group G itself. B) The set containing only the identity element. C) The set of all elements with inverses. D) The set of all generators of G.

12. Which of the following is NOT a subgroup of the group of integers under addition (Z, +)?

A) The set of even integers (2Z, +) B) The set of odd integers (2Z+1, +) C) The set of multiples of 5 (5Z, +) D) The set containing only zero ({0}, +)

13. What is a homomorphism between two groups (G, *) and (G', ∘)?

A) A bijective function f: G -> G' such that f(a) ∘ f(b) = f(a * b). B) A function f: G -> G' such that f(a * b) = f(a) ∘ f(b) for all a, b in G. C) A function f: G -> G' such that f(a * b) = f(a) + f(b). D) A function f: G -> G' such that f(a) * f(b) = f(a ∘ b).

14. A homomorphism f: G -> G' is called an isomorphism if it is also:

A) Injective B) Surjective C) Bijective D) Constant

15. Two groups G and G' are said to be isomorphic if:

A) There exists a surjective homomorphism from G to G'. B) There exists an injective homomorphism from G to G'. C) There exists a bijective homomorphism (isomorphism) between them. D) They have the same number of elements.

16. What does it mean for two groups to be isomorphic?

A) They are structurally identical, differing only in the notation of their elements and operation. B) They are subsets of each other. C) They share the same identity element. D) They are both commutative.

17. If f: G -> G' is a group homomorphism, what is the image of the identity element of G under f?

A) The identity element of G'. B) The identity element of G. C) Any element of G'. D) The inverse of f(e).

18. If f: G -> G' is a group homomorphism, what is the relationship between f(a⁻¹) and (f(a))⁻¹?

A) f(a⁻¹) = f(a). B) f(a⁻¹) = (f(a))⁻¹. C) f(a⁻¹) = -f(a). D) f(a⁻¹) is not related to (f(a))⁻¹.

19. The kernel of a group homomorphism f: G -> G' is defined as:

A) The set of all elements in G' that are images of elements in G. B) The set of all elements in G whose image under f is the identity element of G'. C) The set of all elements in G that map to themselves. D) The set of all elements in G' that have preimages in G.

20. What is a group representation?

A) A mapping from a group G to the set of all its subgroups. B) A mapping from a group G to the set of invertible matrices of a certain size, preserving the group structure. C) A function that maps group elements to scalars. D) A way to decompose a group into its constituent elements.

21. In the context of group representations, what does it mean for a mapping D: G -> GL(n, C) to be a representation?

A) D(g₁g₂) = D(g₁) + D(g₂) for all g₁, g₂ in G. B) D(g₁g₂) = D(g₁)D(g₂) for all g₁, g₂ in G, where D(g) are n x n matrices. C) D(g₁g₂) = D(g₁) - D(g₂) for all g₁, g₂ in G. D) D(g₁g₂) = D(g₁) / D(g₂) for all g₁, g₂ in G.

22. What is the set GL(n, C) in the context of group representations?

A) The set of all n x n complex matrices. B) The set of all invertible n x n complex matrices. C) The set of all n x n real matrices. D) The set of all diagonal n x n complex matrices.

23. A representation D is called 'faithful' if:

A) The kernel of the representation is the entire group G. B) The kernel of the representation is only the identity element. C) The representation is a homomorphism but not necessarily injective. D) The matrices involved are always diagonal.

24. What is the character of a group representation D?

A) The matrix D(g) itself for each group element g. B) The determinant of the matrix D(g). C) The trace of the matrix D(g). D) The inverse of the matrix D(g).

25. A representation D is equivalent to another representation D' if:

A) D(g) = D'(g) for all g in G. B) There exists an invertible matrix S such that D'(g) = S⁻¹ D(g) S for all g in G. C) The characters of D and D' are different. D) They are representations of different groups.

26. What is an irreducible representation?

A) A representation that cannot be decomposed into a direct sum of smaller representations. B) A representation where all matrices are diagonal. C) A representation where the group is abelian. D) A representation that is not faithful.

27. A representation D is reducible if:

A) It is the trivial representation. B) There exists a non-trivial invariant subspace under the action of the group. C) The group G is infinite. D) The character is always zero.

28. The theorem regarding the decomposition of a representation states that any finite-dimensional representation of a finite group (over the complex numbers) can be uniquely decomposed into a direct sum of:

A) Irreducible representations. B) Trivial representations. C) Faithful representations. D) Commutative representations.

29. What is a unitary representation?

A) A representation where all matrices are real. B) A representation where D(g) is a unitary matrix for every g in G. C) A representation where the character is always 1. D) A representation where the group G is finite.

30. A matrix U is unitary if:

A) U = Uᵀ (transpose). B) U = U* (conjugate transpose). C) U*U = I (identity matrix). D) det(U) = 1.

31. Why are unitary representations particularly important in physics?

A) They always correspond to finite groups. B) They preserve the inner product, ensuring probabilities remain constant in quantum mechanics. C) They are easier to compute than non-unitary representations. D) They are only applicable to abelian groups.

32. If D is a unitary representation, what is the relationship between D(g) and D(g)⁻¹?

A) D(g)⁻¹ = D(g). B) D(g)⁻¹ = D(g)ᵀ. C) D(g)⁻¹ = D(g)* (conjugate transpose). D) D(g)⁻¹ = -D(g).

33. The set of all n x n unitary matrices forms a group under matrix multiplication. This group is denoted as:

A) GL(n, C) B) SL(n, C) C) U(n) D) SO(n)

34. If D is a representation and H is a normal subgroup of G, the quotient group G/H can have its own representation induced from D. This is related to:

A) Isomorphism theorems B) Direct product decomposition C) Irreducible representations D) Commutativity

35. Which of the following is NOT a property of the character χ of a representation D?

A) χ(e) is equal to the dimension of the representation. B) χ(g⁻¹) = χ(g). C) χ(ghg⁻¹) = χ(h) for any g, h in G. D) If D₁ and D₂ are representations, then χ(D₁ ⊕ D₂) = χ(D₁) + χ(D₂).

36. What is the dimension of the trivial representation of any group G?

A) The order of the group G. B) 1 C) The number of elements in the center of G. D) 0

37. Consider the group S₃ (symmetric group on 3 elements) under composition. Which statement about its irreducible representations is true?

A) It has only one irreducible representation (the trivial one). B) It has exactly three irreducible representations. C) It has infinitely many irreducible representations. D) It has two irreducible representations, one of dimension 1 and one of dimension 2.

38. The number of non-equivalent irreducible representations of a finite group G is equal to:

A) The order of the group G. B) The number of elements in the center of G. C) The number of conjugacy classes of G. D) The dimension of the group.

39. What is the relationship between isomorphism and group representations?

A) Isomorphic groups have completely different representation theories. B) Isomorphic groups have equivalent representation theories, meaning they have the same number and dimensions of irreducible representations. C) Only abelian groups can be isomorphic. D) Representations are only defined for non-isomorphic groups.

40. If D is a representation of G, and H is a subgroup of G, then the restriction of D to H, denoted Res<0xE1><0xB5><0x8F> D, is:

A) A representation of G/H. B) A representation of H. C) Always an irreducible representation. D) A homomorphism from G to GL(n, C).

41. The concept of induced representations is used to construct representations of a group G from representations of:

A) A larger group containing G. B) A subgroup H of G. C) The center of G. D) The trivial subgroup.

42. Which property must a subset H of a group G satisfy to be a subgroup, besides being non-empty?

A) Closure under the group operation. B) Existence of an identity element within H. C) Existence of inverses for all elements within H. D) All of the above.

43. If f: G -> G' is an isomorphism, then the kernel of f is:

A) The entire group G. B) The identity element of G. C) The set of all elements in G that map to the identity in G'. D) The set of all elements in G'.

44. A group representation D: G -> GL(n, C) is called reducible if:

A) There exists a proper non-trivial subspace of Cⁿ that is invariant under D(g) for all g ∈ G. B) The group G is abelian. C) The character of the representation is identically zero. D) The representation is faithful.

45. The set of all 1x1 unitary matrices forms which group?

A) SO(1) B) U(1) C) SU(1) D) O(1)

46. What is the primary advantage of using characters instead of the full representation matrices?

A) Characters are always complex numbers, simplifying calculations. B) Characters are invariant under change of basis, making them unique for equivalent representations. C) Characters reveal the group structure more directly than matrices. D) Characters are always integers.

47. If a group G has N irreducible representations, and the dimensions of these representations are d₁, d₂, ..., d<0xE2><0x82><0x99>, then which relation holds?

A) Σᵢ<0xE1><0xB5><0x83>₁<0xE1><0xB5><0x83> dᵢ = |G| (order of the group) B) Σᵢ<0xE1><0xB5><0x83>₁<0xE1><0xB5><0x83> dᵢ² = |G| (order of the group) C) Σᵢ<0xE1><0xB5><0x83>₁<0xE1><0xB5><0x83> dᵢ² = N D) Σᵢ<0xE1><0xB5><0x83>₁<0xE1><0xB5><0x83> dᵢ = N

48. The condition for a mapping f: G -> G' to be a homomorphism is:

A) f(a * b) = f(a) + f(b) B) f(a * b) = f(a) * f(b) C) f(a * b) = f(a) ∘ f(b) D) f(a * b) = f(b) * f(a)

49. What is the defining characteristic of a subgroup H of G using the one-step subgroup test?

A) H is non-empty and for all a, b in H, a * b is in H. B) H is non-empty and for all a in H, a⁻¹ is in H. C) H is non-empty and for all a, b in H, a * b⁻¹ is in H. D) H is non-empty and contains the identity element.

Definition of groups - subgroups, homomorphism and isomorphism, group representations, irreducible representations, unitary representations - Online Test

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