TRB BT Assistant - Lagrange Theorem and Counting Principles Online Test (MCQ Quiz)

1. What is the statement of Lagrange's Theorem regarding the order of a subgroup?

A) The order of an element in a group divides the order of the group. B) The order of a subgroup divides the order of the group. C) The order of a group is always prime. D) The order of the product of two elements is the product of their orders.

2. If G is a finite group and H is a subgroup of G, then Lagrange's Theorem states that:

A) |H| = |G| B) |G| divides |H| C) |H| divides |G| D) |H| is a prime number

3. Consider the group of integers modulo 12 under addition, Z_12. What is the order of the subgroup generated by 3, which is {0, 3, 6, 9}?

A) 3 B) 4 C) 6 D) 12

4. In the context of Lagrange's Theorem, what is the relationship between the order of an element 'a' in a group G and the order of the group G?

A) The order of 'a' is equal to the order of G. B) The order of G divides the order of 'a'. C) The order of 'a' divides the order of G. D) The order of 'a' is always a prime number.

5. What is the index of a subgroup H in a group G, denoted by [G:H]?

A) The order of H B) The order of G C) The number of distinct left (or right) cosets of H in G D) The number of elements in the intersection of H with its conjugates

6. Lagrange's Theorem also implies a relationship between the order of a group, the order of a subgroup, and the index of the subgroup. What is this relationship?

A) |G| = |H| * [G:H] B) |H| = |G| * [G:H] C) [G:H] = |G| * |H| D) |G| + |H| = [G:H]

7. If a group G has order 10, which of the following cannot be the order of a subgroup of G?

A) 1 B) 2 C) 5 D) 7

8. What is a necessary condition for a subgroup H to be a normal subgroup of G, related to cosets?

A) Every left coset is equal to the corresponding right coset. B) The order of H must be equal to the order of G. C) The index [G:H] must be 1. D) H must contain only the identity element.

9. What is the converse of Lagrange's Theorem?

A) If a number 'n' divides the order of a group G, then G has a subgroup of order 'n'. B) If a number 'n' divides the order of a group G, then G has an element of order 'n'. C) If a group G has an element of order 'n', then 'n' divides the order of G. D) If a group G has a subgroup of order 'n', then 'n' divides the order of G.

10. Which theorem is a generalization of Lagrange's Theorem for infinite groups?

A) Cauchy's Theorem B) Sylow's First Theorem C) The Fundamental Theorem of Finite Abelian Groups D) The Jordan-Hölder Theorem

11. Consider the symmetric group S_3. What is its order?

A) 3 B) 4 C) 6 D) 12

12. Which of the following is a subgroup of S_3?

A) A cyclic group of order 4 B) A cyclic group of order 3 C) A cyclic group of order 5 D) A cyclic group of order 6

13. What is the order of the element (1 2 3) in the symmetric group S_3?

A) 1 B) 2 C) 3 D) 6

14. The set of all permutations of n distinct objects forms which algebraic structure?

A) A cyclic group B) A non-abelian group C) An abelian group D) A ring

15. What is the order of the alternating group A_4?

A) 12 B) 24 C) 6 D) 8

16. If G is a group of prime order p, then G is isomorphic to which group?

A) The cyclic group Z_p B) The cyclic group Z_p^2 C) The symmetric group S_p D) The alternating group A_p

17. What is the fundamental counting principle (also known as the multiplication principle)?

A) If an event can occur in 'm' ways and another independent event can occur in 'n' ways, then the two events can occur in sequence in m+n ways. B) If an event can occur in 'm' ways and another independent event can occur in 'n' ways, then the two events can occur in sequence in m*n ways. C) If an event can occur in 'm' ways and another independent event can occur in 'n' ways, then the two events can occur in sequence in m-n ways. D) If an event can occur in 'm' ways and another independent event can occur in 'n' ways, then the two events can occur in sequence in m/n ways.

18. If there are 'm' ways to do one thing and 'n' ways to do another, and these choices are independent, how many ways are there to do both?

A) m + n B) m - n C) m * n D) m / n

19. Consider forming a 3-digit number using digits 1, 2, 3, 4, 5 without repetition. How many such numbers can be formed?

A) 5 * 5 * 5 B) 5 * 4 * 3 C) 5 + 4 + 3 D) 5!

20. What does the addition principle (or addition rule) state?

A) If a task can be done in 'm' ways and another task can be done in 'n' ways, and the tasks cannot be done at the same time, then there are m*n ways to do either task. B) If a task can be done in 'm' ways and another task can be done in 'n' ways, and the tasks cannot be done at the same time, then there are m+n ways to do either task. C) If a task can be done in 'm' ways and another task can be done in 'n' ways, and the tasks can be done at the same time, then there are m+n ways to do either task. D) If a task can be done in 'm' ways and another task can be done in 'n' ways, and the tasks can be done at the same time, then there are m*n ways to do either task.

21. If there are 3 routes from city A to city B, and 4 routes from city B to city C, how many distinct ways are there to travel from city A to city C via city B?

A) 3 + 4 = 7 B) 4 - 3 = 1 C) 3 * 4 = 12 D) 4 / 3

22. A committee of 3 people is to be selected from a group of 5 people. How many different committees can be formed?

A) 5 * 4 * 3 B) 5 + 4 + 3 C) 5 choose 3 (C(5,3)) D) 5!

23. What is the formula for combinations, 'n choose k' (C(n,k))?

A) n! / (k! * (n-k)!) B) n! / k! C) n! / (n-k)! D) (n-k)! / n!

24. What is the formula for permutations, 'n permute k' (P(n,k))?

A) n! / (k! * (n-k)!) B) n! / k! C) n! / (n-k)! D) (n-k)! / n!

25. In how many ways can the letters of the word 'MATH' be arranged?

A) 4 B) 8 C) 12 D) 24

26. How many distinct permutations are there of the letters in the word 'APPLE'?

A) 5! B) 5! / 2! C) 5! / 3! D) 5! / 2!3!

27. If a set has n elements, how many subsets does it have?

A) n B) 2n C) 2^n D) n^2

28. How many ways can you arrange 5 distinct books on a shelf?

A) 5 B) 10 C) 25 D) 120

29. Consider the group Z_6 under addition modulo 6. What is the order of the element 4?

A) 1 B) 2 C) 3 D) 6

30. Which of the following is NOT necessarily true for any finite group G?

A) G has a subgroup of order 1. B) G has a subgroup of order |G|. C) G has an element of order |G|. D) The order of every element of G divides |G|.

31. If G is a group and H is a subgroup, and g is an element of G, what is a left coset of H in G?

A) {hg | h in H} B) {gh | h in H} C) {g * h | h in H} D) {h * g | h in H}

32. Lagrange's Theorem is a fundamental result in which area of mathematics?

A) Number Theory B) Abstract Algebra C) Linear Algebra D) Real Analysis

33. If G is a group of order 12, which of the following is a possible order for an element in G?

A) 7 B) 8 C) 10 D) 4

34. What is the order of the group of integers modulo n under addition, Z_n?

A) 1 B) n C) n-1 D) infinity

35. Consider the group of units modulo 5, U(5) = {1, 2, 3, 4} under multiplication modulo 5. What is the order of the subgroup generated by 2?

A) 1 B) 2 C) 3 D) 4

36. If H is a subgroup of G, and |G| = 24 and |H| = 6, what is the index [G:H]?

A) 2 B) 3 C) 4 D) 18

37. What is the number of permutations of n distinct objects taken k at a time?

A) C(n,k) B) P(n,k) C) n^k D) k^n

38. In how many ways can a president, vice-president, and treasurer be selected from a club of 10 members?

A) C(10,3) B) P(10,3) C) 10^3 D) 10+3

39. How many binary strings of length 8 are there?

A) 8 B) 16 C) 2^8 D) 8^2

40. If you have 5 different flavors of ice cream and you want to choose 2 different flavors for a cone, how many combinations are possible?

A) P(5,2) B) C(5,2) C) 5+2 D) 5*2

41. What is the total number of ways to arrange the letters in the word 'MISSISSIPPI'?

A) 11! B) 11! / (4! * 4! * 2!) C) 11! / (4! * 4! * 2! * 1!) D) 11! / (4! * 4!)

42. Which counting principle is used when calculating the number of possible outcomes when rolling two distinct dice?

A) Addition Principle B) Multiplication Principle C) Inclusion-Exclusion Principle D) Pigeonhole Principle

43. A password must be 4 characters long and can consist of uppercase letters or digits. How many possible passwords are there?

A) 26 * 10 B) (26+10)^4 C) 26^4 * 10^4 D) 4^26 * 4^10

44. If a group G has order 30, what are the possible orders of its subgroups?

A) Any divisor of 30 B) Only prime divisors of 30 C) Only divisors that are powers of primes D) Only 1, 2, 3, 5, 6, 10, 15, 30

45. What does Cauchy's Theorem state for finite groups?

A) If G is a finite group and p is a prime dividing |G|, then G has an element of order p. B) If G is a finite group and p is a prime dividing |G|, then G has a subgroup of order p. C) If G is a finite group and p is a prime dividing |G|, then G is cyclic. D) If G is a finite group and p is a prime dividing |G|, then G has a normal subgroup of order p.

46. Consider the group of symmetries of an equilateral triangle (D_3). What is its order?

A) 3 B) 4 C) 6 D) 12

47. If a subgroup H has index 2 in G, what can be said about H?

A) H is always trivial. B) H is always cyclic. C) H is always a normal subgroup of G. D) H is always abelian.

48. How many non-isomorphic groups of order 4 exist?

A) 1 B) 2 C) 3 D) 4

49. If there are 7 choices for the first letter of a word and 5 choices for the second letter, how many 2-letter words can be formed if repetition is allowed?

A) 7 + 5 B) 7 - 5 C) 7 * 5 D) 7^5

50. What is the primary implication of Lagrange's Theorem for the structure of finite groups?

A) It guarantees the existence of a unique subgroup for every divisor of the group's order. B) It establishes that the order of every element and subgroup must divide the order of the group. C) It proves that all finite groups are cyclic. D) It provides a method for constructing all possible finite groups.

Lagrange Theorem and Counting Principles - Online Test

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