3.3.1 - introduction and Properties of Subgroups.

1. What is the definition of a subgroup of a group G?

(A) A non-empty subset H of G that includes only the identity element
(B) A non-empty subset H of G that is closed under the group operation and includes the identity and inverses
(C) A non-empty subset H of G that contains all elements of G
(D) A non-empty subset H of G that does not include the identity element

Correct Answer: (B) A non-empty subset H of G that is closed under the group operation and includes the identity and inverses.

2. Which of the following is NOT a necessary condition for a subset H of a group G to be a subgroup?

(A) H must be closed under the group operation
(B) Every element of H must have an inverse in H
(C) H must be the same as G
(D) The identity element of H must be the same as that of G

Correct Answer: (C) H must be the same as G.

3. Which of the following theorems provides a necessary and sufficient condition for a subset H of a group G to be a subgroup?

(A) The subset H must be closed under composition and contain the identity element
(B) For any a ∈ H and b ∈ H, a o b ∈ H and a−1 ∈ H
(C) For any a ∈ H and b ∈ H, a o b−1 ∈ H
(D) H must include all elements of G

Correct Answer: (B) For any a ∈ H and b ∈ H, a o b ∈ H and a−1 ∈ H.

4. Which of the following is an example of a proper subgroup?

(A) The group G itself
(B) The trivial subgroup consisting only of the identity element
(C) The additive group of integers within the additive group of rational numbers
(D) The group of all non-zero real numbers

Correct Answer: (C) The additive group of integers within the additive group of rational numbers.

5. In the context of subgroups, which property is true regarding the identity element?

(A) The identity of a subgroup is different from the identity of the group
(B) The identity of a subgroup is the same as that of the group
(C) A subgroup does not need to contain an identity element
(D) The identity element of a subgroup is arbitrary

Correct Answer: (B) The identity of a subgroup is the same as that of the group.

6. What is a complex in the context of group theory?

(A) Any subset of a group that is a subgroup
(B) A subset of a group, whether it is a subgroup or not
(C) A subgroup that is not proper
(D) A subgroup containing only the identity element

Correct Answer: (B) A subset of a group, whether it is a subgroup or not.

7. Which theorem states that a subset H of a group G is a subgroup if and only if for any a ∈ H and b ∈ H, a o b−1 ∈ H?

(A) Theorem 1
(B) Theorem 2
(C) Theorem of Closure
(D) Theorem of Identity

Correct Answer: (B) Theorem 2.

8. What does the closure axiom ensure in the context of subgroups?

(A) Every element of H has an inverse in H
(B) The subgroup contains the identity element
(C) The subset H is closed under the group operation
(D) The subgroup H is equal to the original group G

Correct Answer: (C) The subset H is closed under the group operation.

9. What is true about the order of elements in a subgroup compared to the original group?

(A) The order of elements in a subgroup is different from the original group
(B) The order of any element of a subgroup is the same as that of the element in the original group
(C) The order of elements in a subgroup is always smaller
(D) The order of elements in a subgroup is arbitrary

Correct Answer: (B) The order of any element of a subgroup is the same as that of the element in the original group.

10. What does associativity in a subgroup imply?

(A) The group operation is associative only within the subgroup
(B) The subgroup has its own associative operation
(C) Since the elements of the subgroup are also elements of the group, the group operation is associative in the subgroup
(D) Associativity is not a requirement for subgroups

Correct Answer: (C) Since the elements of the subgroup are also elements of the group, the group operation is associative in the subgroup.

11. Which condition is necessary for a subset H of a group G to be a subgroup according to Theorem 1?

(A) H must contain the identity element of G
(B) For any a ∈ H and b ∈ H, a o b ∈ H and a−1 ∈ H
(C) For any a ∈ H, a o b−1 ∈ H
(D) H must be closed under the group operation

Correct Answer: (B) For any a ∈ H and b ∈ H, a o b ∈ H and a−1 ∈ H.

12. Which property of a subgroup is guaranteed by the associativity axiom?

(A) The subset H is closed under the group operation
(B) The identity element is present in H
(C) The group operation is associative within H
(D) Every element in H has an inverse in H

Correct Answer: (C) The group operation is associative within H.

13. What does the existence of an inverse in a subgroup guarantee?

(A) Each element of the subgroup has a corresponding inverse within the group
(B) The subgroup must contain all elements of the group
(C) The identity element of the subgroup is different from the identity of the group
(D) Each element of the subgroup has a corresponding inverse in the group

Correct Answer: (A) Each element of the subgroup has a corresponding inverse within the group.

14. In the context of subgroup properties, what is meant by the term "proper subgroup"?

(A) A subgroup that contains the identity element
(B) A subgroup that is not equal to the entire group
(C) A subgroup that is identical to the original group
(D) A trivial subgroup consisting only of the identity element

Correct Answer: (B) A subgroup that is not equal to the entire group.

15. What is the relationship between the inverses of elements in a subgroup and the group?

(A) The inverse of an element in a subgroup is different from its inverse in the group
(B) The inverse of an element in a subgroup is the same as its inverse in the group
(C) The inverse of an element in a subgroup does not exist
(D) The inverse of an element in a subgroup is arbitrary

Correct Answer: (B) The inverse of an element in a subgroup is the same as its inverse in the group.

3.3.2 - Order of an Element of a Group.

16. What is the definition of the order of an element a in a group G?

(A) The smallest integer n such that aⁿ = e
(B) The largest integer n such that aⁿ = e
(C) The number of elements in the group G
(D) The number of distinct elements generated by a

Correct Answer: (A) The smallest integer n such that aⁿ = e.

17. If no positive integer n exists such that aⁿ = e, what is the order of a?

(A) Zero order
(B) Finite order
(C) Infinite order
(D) Undefined

Correct Answer: (C) Infinite order.

18. What is the order of the identity element e in a group G?

(A) 0
(B) 1
(C) The order of the group G
(D) Any positive integer

Correct Answer: (B) 1.

19. In a group of order 2, if the elements are A, B, and AB, which of the following statements is true?

(A) AB = BA
(B) A and B are not invertible
(C) A = B
(D) AB is not in the group

Correct Answer: (A) AB = BA.

20. If a has order 3 in a group G, which of the following is true?

(A) a² = e
(B) a³ = e
(C) a⁴ = e
(D) a¹ = e

Correct Answer: (B) a³ = e.

21. In a group G, if a⁴ = e, what could be the possible orders of a?

(A) 1, 2, or 4
(B) 1, 2, 3, or 4
(C) 1, 3, or 4
(D) 2, 3, or 4

Correct Answer: (A) 1, 2, or 4.

22. If an element a in a group G has order 6, which of the following must be true?

(A) a² = e
(B) a³ = e
(C) a⁶ = e
(D) a⁵ = e

Correct Answer: (C) a⁶ = e.

23. What is the order of the product of two elements a and b in a finite group, if both a and b have finite orders?

(A) The order is the product of their individual orders
(B) The order is the least common multiple of their orders
(C) The order is the greatest common divisor of their orders
(D) The order is always equal to 1

Correct Answer: (B) The order is the least common multiple of their orders.

3.3.3 - Cyclic Group and Its Properties.

24. What defines a cyclic group G?

(A) A group where every element is the inverse of another element.
(B) A group where there exists an element a such that every element x ∈ G is of the form aⁿ, where n is an integer.
(C) A group where every element commutes with every other element.
(D) A group where the order of every element is the same.

Correct Answer: (B) A group where there exists an element a such that every element x ∈ G is of the form aⁿ, where n is an integer.

25. If G is a cyclic group generated by a, what is the order of G if the order of a is n?

(A) n²
(B) n
(C) n + 1
(D) n - 1

Correct Answer: (B) n.

26. Which of the following is true about cyclic groups?

(A) Every cyclic group is not necessarily abelian.
(B) The order of a cyclic group is not related to the order of its generator.
(C) The inverse of a generator of a cyclic group is also a generator.
(D) Cyclic groups cannot be finite.

Correct Answer: (C) The inverse of a generator of a cyclic group is also a generator.

27. For a cyclic group G with generator a and order n, which elements are generators of G?

(A) Elements that are multiples of n.
(B) Elements that are relatively prime to n.
(C) Elements that are factors of n.
(D) Only the identity element.

Correct Answer: (B) Elements that are relatively prime to n.

28. In a cyclic group of order 5, which of the following elements are generators?

(A) a¹, a², a³, a⁴
(B) a¹
(C) a², a⁴
(D) a², a³

Correct Answer: (A) a¹, a², a³, a⁴.

29. If a is a generator of a cyclic group G, what is the order of a^-1?

(A) The same as the order of a.
(B) Half of the order of a.
(C) Twice the order of a.
(D) Zero.

Correct Answer: (A) The same as the order of a.

30. If G is a cyclic group generated by a and has an order of n, how many distinct elements does G have?

(A) n - 1
(B) 2ⁿ
(C) n
(D) n²

Correct Answer: (C) n.

31. Which of the following statements is true about the generators of a cyclic group?

(A) The generators of a cyclic group are always the powers of the identity element.
(B) There are exactly φ(n) generators in a cyclic group of order n, where φ is the Euler's totient function.
(C) A cyclic group can have no generators.
(D) A cyclic group’s generators are always the elements of maximum order.

Correct Answer: (B) There are exactly φ(n) generators in a cyclic group of order n, where φ is the Euler's totient function.

32. In a cyclic group G generated by a, if a^k is an element of G, what is the order of a^k?

(A) n / gcd(n, k)
(B) n ⋅ gcd(n, k)
(C) lcm(n, k)
(D) gcd(n, k)

Correct Answer: (A) n / gcd(n, k).