8.03e-h Orders of elements, subgroups, cyclic groups and generators Quiz: OCR Further Maths, Unit 5

The 20 questions

1. What is the order of the identity element in any group?

   • 1
   • 2
   • Infinite
   • 0

2. What is the order of a finite group element a?

   • Largest positive n
   • Number of subgroups
   • Index of element
   • Smallest positive n

3. Which theorem states that the order of a subgroup divides the group order?

   • Euler's theorem
   • Lagrange's theorem
   • Cayley's theorem
   • Fermat's theorem

4. What type of group is generated by a single element?

   • Abelian group
   • Normal group
   • Cyclic group
   • Simple group

5. How many generators does the cyclic group C6 have?

   • 3
   • 6
   • 2
   • 1

6. What is the order of the element 4 in the additive group Z12?

   • 3
   • 6
   • 4
   • 2

7. What is the relationship between the order of an element and its inverse?

   • Inverse is half
   • They are equal
   • Inverse is double
   • Always reciprocal

8. If a finite group has prime order p, what must it be?

   • Trivial
   • Infinite
   • Cyclic
   • Non-abelian

9. According to Lagrange's theorem, which order is possible for a subgroup of order 10?

   • 6
   • 3
   • 5
   • 4

10. What is a subgroup containing only the identity element called?

    • Cyclic subgroup
    • Normal subgroup
    • Trivial subgroup
    • Proper subgroup

11. Which function gives the number of generators of a cyclic group of order n?

    • Gamma function
    • Factorial function
    • Signum function
    • Euler's totient function

12. What structure must every subgroup of a cyclic group possess?

    • Cyclic
    • Symmetric
    • Infinite
    • Non-abelian

13. Which set of elements generates the additive group Z8?

    • 0, 1, 3, 5
    • 2, 4, 6, 8
    • 1, 2, 4, 8
    • 1, 3, 5, 7

14. If element g generates a cyclic group G, which other element must generate G?

    • Inverse of g
    • Identity
    • Cube of g
    • Square of g

15. How many subgroups does the cyclic group C12 have in total?

    • 12
    • 6
    • 4
    • 2

16. For a non-empty finite subset H of a group, which condition ensures H is a subgroup?

    • Infinite order
    • Contains two elements
    • Closed under operation
    • Commutative

17. In a cyclic group of order 15 with generator g, what is the order of g^6?

    • 15
    • 5
    • 6
    • 3

18. What is the order of every non-identity element in the Klein four-group?

    • 1
    • 4
    • 2
    • 3

19. What is a subgroup H of G called if H is not equal to G?

    • Trivial subgroup
    • Abelian subgroup
    • Proper subgroup
    • Infinite subgroup

20. If an element a in a group has order 6, what is the order of subgroup <a>?

    • 3
    • 6
    • 1
    • 12

Related quizzes

• Recurrence relations and properties of sequences Quiz · 8.01a-c · 20 questions
• Fibonacci numbers, solving recurrences and modelling Quiz · 8.01e-i · 20 questions
• Number bases and divisibility Quiz · 8.02a-d · 20 questions
• Finite (modular) arithmetic and prime numbers Quiz · 8.02e-i · 20 questions
• Euclid's lemma, Fermat's little theorem and the binomial theorem Quiz · 8.02k-o · 20 questions
• Binary operations and the definition of a group Quiz · 8.03a-c · 20 questions
• Structure of finite groups, Lagrange's theorem and isomorphism Quiz · 8.03i-m · 20 questions
• Vector product Quiz · 8.04a · 20 questions
• Surfaces, sections, contours and partial derivatives Quiz · 8.05a-d · 20 questions
• Stationary points and tangent planes Quiz · 8.05e-g · 20 questions