8.03i-m Structure of finite groups, Lagrange's theorem and isomorphism Quiz: OCR Further Maths, Unit 5

The 20 questions

1. What does Lagrange's theorem state about the order of a subgroup?

   • Equals group order
   • Exceeds group order
   • Divides group order
   • Is always prime

2. A finite group has order 12. Which of these could be the order of a subgroup?

   • 8
   • 4
   • 7
   • 5

3. What structure must every finite group of prime order have?

   • Cyclic
   • Non-abelian
   • Infinite
   • Dihedral

4. What is a bijective homomorphism between two groups called?

   • Endomorphism
   • Automorphism
   • Epimorphism
   • Isomorphism

5. How many proper non-trivial subgroups does a group of order 7 have?

   • 1
   • 0
   • 2
   • 6

6. A finite group has order 15. What is the maximum possible order of an element?

   • 30
   • 15
   • 5
   • 3

7. What is the order of the smallest non-cyclic group?

   • 4
   • 3
   • 2
   • 6

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

   • 1
   • 4
   • 3
   • 2

9. Any finite cyclic group of order n is isomorphic to which group?

   • Dihedral group Dn
   • Integers modulo n
   • Real numbers
   • Symmetric group Sn

10. In an isomorphism between groups, where does the domain identity map to?

    • Codomain identity
    • Domain identity
    • Zero element
    • Inverse element

11. A group has order 20. Can it contain an element of order 6?

    • No
    • Yes
    • Only if cyclic
    • Only if abelian

12. What is the size of every left coset of a finite subgroup H?

    • Equal to 1
    • Equal to |H|
    • Divisor of |H|
    • Greater than |H|

13. The set of all left cosets of a subgroup H in G forms what?

    • Homomorphism
    • Cyclic group
    • Partition of G
    • Subgroup of G

14. How many generators does a cyclic group of order 8 have?

    • 8
    • 2
    • 6
    • 4

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

    • 4
    • 3
    • 2
    • 1

16. If two groups are isomorphic and one is abelian, what is the other?

    • Infinite
    • Non-abelian
    • Cyclic
    • Abelian

17. What is the order of the smallest non-abelian group?

    • 6
    • 4
    • 3
    • 5

18. If element g generates a finite group G, what is the order of g?

    • Always 1
    • Always prime
    • Order of G
    • Half of order

19. A group G has order 24 and subgroup H has order 6. What is the index [G:H]?

    • 18
    • 4
    • 144
    • 3

20. Group isomorphism is an example of what type of mathematical relation?

    • Partial order
    • Linear relation
    • Equivalence relation
    • Reflexive relation

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