DG9-DG12 Finite and infinite groups, subgroups, Lagrange's theorem and isomorphism Quiz: AQA Further Maths, Unit 5

The 20 questions

1. According to Lagrange's theorem, what must the order of a subgroup divide?

   • Group order
   • Element order
   • Number of generators
   • Group index

2. Which term describes a subset of group G that is also a group under G's operation?

   • Subgroup
   • Coset
   • Generator
   • Quotient group

3. What type of group has all its elements generated by powers of a single element?

   • Dihedral group
   • Symmetric group
   • Cyclic group
   • Abelian group

4. Which property must hold for all elements in an abelian group?

   • Commutativity
   • Distributivity
   • Closure
   • Associativity

5. What is the order of the symmetry group of an equilateral triangle, D3?

   • 6
   • 12
   • 3
   • 9

6. Why is the dihedral group D4 classified as non-abelian?

   • Non-commuting operations
   • No identity element
   • Cyclic structure
   • Infinite order

7. What is the order of the symmetry group of a square, D4?

   • 4
   • 8
   • 6
   • 16

8. Which elements generate the group Z6 under addition modulo 6?

   • 1 and 3
   • 0 and 3
   • 1 and 5
   • 2 and 4

9. Which subgroup order is impossible for a group of order 12?

   • 2
   • 3
   • 4
   • 5

10. What are the possible orders of subgroups of Z6 under addition modulo 6?

    • 1, 3 and 5
    • 1, 2, 4 and 6
    • 1, 2, 3 and 6
    • 2, 3 and 6

11. Which set is a subgroup of Z12 under addition modulo 12?

    • {1, 2, 3, 4}
    • {0, 4, 8, 12}
    • {0, 5, 10}
    • {0, 3, 6, 9}

12. Which of these sets forms an infinite group?

    • Integers modulo 6
    • Integers under addition
    • Symmetries of a square
    • Rotations of a cube

13. Which property prevents ℤ4 and the Klein four-group ℤ2 × ℤ2 from being isomorphic?

    • Group orders differ
    • Element orders differ
    • Identity elements differ
    • Commutativity differs

14. What property prevents ℤ6 and the symmetric group S3 from being isomorphic?

    • Group order
    • Identity element
    • Commutativity
    • Associativity

15. In Z6 under addition modulo 6, what is the order of the element 2?

    • 3
    • 2
    • 6
    • 1

16. What is meant by the order of a group?

    • Number of elements
    • Number of subgroups
    • Highest element order
    • Number of generators

17. By Lagrange's theorem, which order is possible for a subgroup of a group of order 8?

    • 5
    • 3
    • 4
    • 6

18. How many generators does the cyclic group Z5 under addition modulo 5 have?

    • 4
    • 5
    • 1
    • 2

19. What is the logical relationship between cyclic groups and abelian groups?

    • Always equivalent
    • Cyclic implies abelian
    • Mutually exclusive
    • Abelian implies cyclic

20. How many proper non-trivial subgroups can a group of prime order have?

    • 0
    • 2
    • 1
    • Infinitely many

Related quizzes

• Graph language, Eulerian graphs and Euler's formula Quiz · DA1-DA3 · 20 questions
• Planarity, complete and bipartite graphs Quiz · DA4-DA5 · 20 questions
• Trees and isomorphism of graphs Quiz · DA6-DA7 · 20 questions
• Network language, spanning trees and route inspection Quiz · DB1-DB3 · 20 questions
• Travelling salesperson bounds and refining network models Quiz · DB4-DB5 · 20 questions
• Flows, cuts and the maximum flow-minimum cut theorem Quiz · DC1-DC3 · 20 questions
• Supersources, augmenting flows and capacities Quiz · DC4-DC7 · 20 questions
• Formulating and solving linear programmes graphically Quiz · DD1-DD2 · 20 questions
• The Simplex algorithm Quiz · DD3-DD4 · 20 questions
• Activity networks and critical paths Quiz · DE1-DE3 · 20 questions