3.2.1 - Introduction to Group

1. What is the closure axiom in a group (G, o)?

A) For all 𝑎 , 𝑏 ∈ 𝐺, 𝑎 ⋅ 𝑏 ∈ 𝐺
B) For all 𝑎 , 𝑏 ∈ 𝐺, 𝑎 ⋅ 𝑏 ∉ 𝐺
C) For some 𝑎 , 𝑏 ∈ 𝐺, 𝑎 ⋅ 𝑏 ∈ 𝐺
D) For no 𝑎 , 𝑏 ∈ 𝐺, 𝑎 ⋅ 𝑏 ∈ 𝐺

Correct Answer: A) For all 𝑎 , 𝑏 ∈ 𝐺, 𝑎 ⋅ 𝑏 ∈ 𝐺

2. Which axiom states that the binary operation is associative?

A) Closure axiom
B) Associative axiom
C) Identity axiom
D) Inverse axiom

Correct Answer: B) Associative axiom

3. What does the identity axiom in a group state?

A) There exists an element 𝑒 ∈ 𝐺 such that 𝑒 ⋅ 𝑎 = 𝑎 ⋅ 𝑒 = 𝑎 for all 𝑎 ∈ 𝐺
B) There is no identity element in G
C) Each element 𝑎 ∈ 𝐺 must be its own inverse
D) The operation is not associative

Correct Answer: A) There exists an element 𝑒 ∈ 𝐺 such that 𝑒 ⋅ 𝑎 = 𝑎 ⋅ 𝑒 = 𝑎 for all 𝑎 ∈ 𝐺

4. What is true about the inverse axiom in a group?

A) Each element of G does not possess an inverse
B) For each element 𝑎 ∈ 𝐺, there exists an element 𝑏 ∈ 𝐺 such that 𝑎 ⋅ 𝑏 = 𝑒 = 𝑏 ⋅ 𝑎
C) Each element 𝑎 ∈ 𝐺 has multiple inverses
D) The inverse of an element is not in G

Correct Answer: B) For each element 𝑎 ∈ 𝐺, there exists an element 𝑏 ∈ 𝐺 such that 𝑎 ⋅ 𝑏 = 𝑒 = 𝑏 ⋅ 𝑎

5. In a group (G, o), how is the identity element 𝑒 characterized?

A) It is an element that, when combined with any other element, results in the identity element itself
B) It is an element that does not interact with any other element
C) It is an element that only exists for some elements in G
D) It is an element that does not belong to G

Correct Answer: A) It is an element that, when combined with any other element, results in the identity element itself

6. Which axiom requires that the operation 𝑜 be associative for all elements of G?

A) Closure axiom
B) Associative axiom
C) Identity axiom
D) Inverse axiom

Correct Answer: B) Associative axiom

7. What does the inverse axiom ensure about each element in a group (G, o)?

A) Each element must have a unique inverse
B) Each element must be its own inverse
C) Each element must have at least one inverse
D) Each element does not need an inverse

Correct Answer: A) Each element must have a unique inverse

8. In a group (G, o), if an element 𝑎 has an inverse 𝑎⁻¹, which of the following is true?

A) 𝑎 ⋅ 𝑎⁻¹ = 𝑒 and 𝑎⁻¹ ⋅ 𝑎 = 𝑒
B) 𝑎 ⋅ 𝑎⁻¹ ≠ 𝑒 and 𝑎⁻¹ ⋅ 𝑎 ≠ 𝑒
C) 𝑎 ⋅ 𝑎⁻¹ = 𝑎 and 𝑎⁻¹ ⋅ 𝑎 = 𝑎⁻¹
D) 𝑎 ⋅ 𝑎⁻¹ = 0 and 𝑎⁻¹ ⋅ 𝑎 = 0

Correct Answer: A) 𝑎 ⋅ 𝑎⁻¹ = 𝑒 and 𝑎⁻¹ ⋅ 𝑎 = 𝑒

9. What is the role of the identity element 𝑒 in a group (G, o)?

A) It acts as a neutral element for the operation 𝑜
B) It must be an element that does not belong to G
C) It must be an element that changes the result of the operation
D) It must be an element that is not associated with any other elements

Correct Answer: A) It acts as a neutral element for the operation 𝑜

10. If an operation 𝑜 in a set 𝐺 is not associative, which of the following is true?

A) 𝐺 cannot be a group
B) 𝐺 is still a group
C) 𝐺 is a commutative group
D) 𝐺 is a cyclic group

Correct Answer: A) 𝐺 cannot be a group

3.2.2 - Groupoid, Semigroup, Monoid and Quanternion Group.

11. Which algebraic system is called a groupoid?

A) A set with a binary operation that is associative and has an identity element
B) A set with a binary operation that is closed under the operation
C) A set with a binary operation that satisfies the associative law
D) A set with a binary operation that has an identity element and is closed under the operation

Correct Answer: B) A set with a binary operation that is closed under the operation

12. What additional property does a semigroup have compared to a groupoid?

A) The presence of an identity element
B) The operation must be associative
C) The operation must be commutative
D) The operation must be distributive

Correct Answer: B) The operation must be associative

13. Which of the following systems is not a semigroup?

A) The set of integers with addition
B) The set of natural numbers with multiplication
C) The set of integers with subtraction
D) The set of real numbers with addition

Correct Answer: C) The set of integers with subtraction

14. What is a monoid?

A) A groupoid with associative operation and an identity element
B) A semigroup with an identity element
C) A groupoid with a commutative operation
D) A semigroup without any additional properties

Correct Answer: B) A semigroup with an identity element

15. Which of the following is an example of a monoid?

A) The set of odd integers with addition
B) The set of integers with subtraction
C) The set of integers with addition
D) The set of real numbers with subtraction

Correct Answer: C) The set of integers with addition

16. In the quaternion group, what is the result of the multiplication i⋅j?

A) −k
B) k
C) i
D) −i

Correct Answer: B) k

17. Which of the following is a property of the quaternion group?

A) The product of any two elements is commutative
B) The set includes only positive elements
C) The product of the elements i, j, and k follows specific rules
D) The set includes only even integers

Correct Answer: C) The product of the elements i, j, and k follows specific rules

18. Which statement is true about the system (I, +) where I is the set of integers?

A) It is a groupoid but not a semigroup
B) It is a semigroup and a monoid
C) It is a semigroup but not a monoid
D) It is neither a groupoid nor a semigroup

Correct Answer: B) It is a semigroup and a monoid

19. What is the result of jk in the quaternion group?

A) i
B) −i
C) −k
D) k

Correct Answer: A) i

20. Which set with a binary operation is an example of a quaternion group?

A) {1,−1,i,−i,j,−j,k,−k} with the specified multiplication rules
B) {0,1,2,3} with addition
C) {2,4,6,8} with multiplication
D) {a,b,c} with a commutative binary operation

Correct Answer: A) {1,−1,i,−i,j,−j,k,−k} with the specified multiplication rules

3.2.3 - Ablelian Group, Finite and Infinite Groups.

21. Which of the following is an example of an abelian group?

A) The set of all odd integers under addition
B) The set of all even integers under addition
C) The set of all non-zero integers under multiplication
D) The set of all positive rational numbers under addition

Correct Answer: B) The set of all even integers under addition

22. Which axiom is NOT satisfied by the set of all even integers under addition?

A) Closure
B) Associativity
C) Identity element
D) Commutativity

Correct Answer: None of the above

23. The set of all non-zero rational numbers under multiplication is a group because it satisfies which of the following?

A) Closure
B) Associativity
C) Identity element
D) All of the above

Correct Answer: D) All of the above

24. Which of the following is an example of a finite group?

A) The set of all integers under addition
B) The set of all positive real numbers under multiplication
C) The set of months in a year under addition
D) The set of all whole numbers under addition

Correct Answer: C) The set of months in a year under addition

25. Which group is considered infinite?

A) The set of all even integers under addition
B) The set {1, -1} under multiplication
C) The set of all whole numbers under addition
D) The set of all non-zero rational numbers under multiplication

Correct Answer: C) The set of all whole numbers under addition

26. Which of the following sets is finite under multiplication?

A) The set of all positive integers
B) The set {1, -1}
C) The set of all real numbers
D) The set of all complex numbers

Correct Answer: B) The set {1, -1}

27. What is the key characteristic of an infinite group?

A) It has a countable number of elements
B) Its elements can be represented by an ellipse
C) It can be listed in roster form
D) It has a finite number of distinct elements

Correct Answer: B) Its elements can be represented by an ellipse

28. Which example illustrates an infinite group?

A) The set {0, 1, 2, 3, 4}
B) The set of all integers under addition
C) The set of all non-zero rational numbers under addition
D) The set {1, ω, ω²}

Correct Answer: B) The set of all integers under addition

29. The set of all non-zero rational numbers under addition does NOT form a group because it lacks which property?

A) Closure
B) Associativity
C) Identity element
D) Inverse elements

Correct Answer: D) Inverse elements

30. Which of the following sets cannot be finite?

A) The set of all months in a year
B) The set of all even integers under addition
C) The set of all whole numbers
D) The set {1, -1, i, -i}

Correct Answer: C) The set of all whole numbers

3.2.4 - Order of a Group

31. What is the order of a finite group?

A) Infinite
B) The number of elements in the group
C) The number of operations possible in the group
D) The size of the group's largest subset

Correct Answer: B) The number of elements in the group

32. How is the order of a group typically denoted?

A) G
B) G*
C) |G|
D) card(G)

Correct Answer: C) |G|

33. What is the order of the symmetric group (S₃, ∘)?

A) 3
B) 6
C) 9
D) 12

Correct Answer: B) 6

34. What is the order of the group (Z, +)?

A) 0
B) 1
C) 6
D) Infinite

Correct Answer: D) Infinite

35. Which of the following groups has a finite order?

A) (Z, +)
B) (S₃, ∘)
C) (R, +)
D) (Q, +)

Correct Answer: B) (S₃, ∘)

36. What is the cardinality of a group of order n?

A) The number of elements in the group
B) The number of distinct subgroups
C) The number of group operations
D) The number of elements in the group’s largest subset

Correct Answer: A) The number of elements in the group

37. In group theory, what does an infinite order group refer to?

A) A group with a finite number of elements
B) A group where the number of elements is uncountably infinite
C) A group where the number of elements is countably infinite
D) A group with exactly 10 elements

Correct Answer: C) A group where the number of elements is countably infinite

38. Which notation correctly represents the order of a group G?

A) size(G)
B) |G|
C) card(G)
D) count(G)

Correct Answer: B) |G|

39. The order of the group (Zₙ, +) is:

A) n
B) n²
C) n+1
D) Infinite

Correct Answer: A) n

40. Which of the following groups has an infinite order?

A) (S₄, ∘)
B) (Zₙ, +)
C) (Z, +)
D) (Z₂, +)

Correct Answer: C) (Z, +)

3.2.5 - Properties of a Group.

41. What does Theorem 1 state about the identity element of a group?

(A) The identity element is a unique element.
(B) The identity element can have multiple values.
(C) The identity element is not required to be unique.
(D) The identity element is always equal to the inverse element.

Correct Answer: (A) The identity element is a unique element.

42. According to the proof of Theorem 1, what happens when you apply the operation to two identity elements e and e′?

(A) e and e′ can be different elements.
(B) e and e′ must be equal.
(C) e and e′ cancel each other out.
(D) e and e′ are not related.

Correct Answer: (B) e and e′ must be equal.

43. What does Theorem 2 state about the inverses in a group?

(A) Each element in a group has no inverse.
(B) Each element in a group has multiple inverses.
(C) Each element in a group has a unique inverse.
(D) The inverse of an element is not guaranteed to be unique.

Correct Answer: (C) Each element in a group has a unique inverse.

44. If a has two inverses a^-1 and a′, what does Theorem 2's proof imply?

(A) a^-1 and a′ can be different elements.
(B) a^-1 and a′ must be equal.
(C) a^-1 and a′ are unrelated.
(D) a^-1 and a′ are not valid inverses.

Correct Answer: (B) a^-1 and a′ must be equal.

45. In the context of Theorem 2, what is true about the uniqueness of the inverse element a^-1 for any element a in a group G?

(A) There can be multiple elements that serve as a^-1.
(B) a^-1 is unique for each element a.
(C) a^-1 does not exist for every a.
(D) The uniqueness of a^-1 is not guaranteed.

Correct Answer: (B) a^-1 is unique for each element a.

46. What does the proof of Theorem 1 imply about the identity element e and e′?

(A) If e is an identity element, then e′ must also be an identity element.
(B) e and e′ can be different as long as they satisfy the identity property.
(C) The identity elements e and e′ need not be equal.
(D) The identity element e is dependent on the operation used in the group.

Correct Answer: (A) If e is an identity element, then e′ must also be an identity element.

47. Which property is essential for proving the uniqueness of the identity element in a group?

(A) The identity element must commute with every element of the group.
(B) The identity element must be different for different operations.
(C) Applying the group operation to two identity elements results in the first identity element.
(D) The identity element must be non-existent in some groups.

Correct Answer: (C) Applying the group operation to two identity elements results in the first identity element.

48. According to the proof of Theorem 2, what is the requirement for two elements to be considered as the inverses of a?

(A) Both elements must satisfy the equation a ⋅ a^-1 = e.
(B) Both elements must satisfy the equation a ⋅ a′ = e.
(C) One element must be the identity element and the other its inverse.
(D) Both elements must be distinct but still valid inverses of a.

Correct Answer: (B) Both elements must satisfy the equation a ⋅ a′ = e.