Lesson DG9-DG12

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

20 questions

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Lesson DG9-DG12, Finite and infinite groups, subgroups, Lagrange's theorem and isomorphism: 20 multiple choice questions for the AQA Further Maths (7367), Unit 5: Optional application 3: discrete mathematics, written with Revision Ninja.

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The 20 questions

  1. What does Lagrange's theorem state for a finite group?

    • The order of any subgroup of a finite group divides the order of the group
    • The order of any element divides the number of subgroups of the group
    • A finite group always has an even number of subgroups
    • The order of any subgroup of a finite group equals the order of the group

  2. What is a subgroup of a group G?

    • A subset of G that is itself a group under the same operation as G
    • Any subset of G that contains the identity element only
    • A group whose order is larger than the order of G
    • A set of elements of G that all have order 2

  3. What is a cyclic group?

    • A group in which the operation is commutative but not associative
    • A group with exactly two elements
    • A group in which every element is a power of a single element, called a generator
    • A group in which every element is its own inverse

  4. What is an abelian group?

    • A group in which the operation is commutative
    • A group that is closed under addition only
    • A group with an even number of elements
    • A group in which every element is its own inverse

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

    • 6
    • 12
    • 3
    • 9

  6. Is the symmetry group of a square, D4, abelian?

    • No, since a rotation and a reflection of the square do not commute in general
    • Yes, since a square has four equal sides
    • No, since the group has only four elements
    • Yes, because every symmetry of a square is a rotation

  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. A group has order 12. Could it have a subgroup of order 5?

    • Yes, since 5 is a prime less than 12
    • Yes, because subgroups can have any order
    • Yes, provided the group is abelian
    • No, since 5 does not divide 12, so Lagrange's theorem rules this out

  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 the following is an infinite group?

    • The symmetry group of a regular hexagon
    • The integers under addition
    • Z5 under addition modulo 5
    • The set {1, 2, 3, 4} under multiplication modulo 5

  13. Are the cyclic group Z4 and the Klein four-group Z2 × Z2 isomorphic?

    • Yes, since both have four elements
    • No, since Z4 has an element of order 4 but Z2 × Z2 has no element of order 4
    • No, since Z4 is not closed under addition
    • Yes, since both are abelian groups of order 4 and so must be isomorphic

  14. Is the cyclic group of order 6 isomorphic to the symmetric group S3, which has order 6?

    • Yes, since both have order 6
    • No, since S3 has only two elements
    • No, since Z6 is abelian but S3 is non-abelian
    • Yes, because both groups are generated by a single element

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

    • 3
    • 2
    • 6
    • 1

  16. What is the order of a group?

    • The number of elements in the group
    • The number of subgroups that the group contains
    • The largest order of any element in the group
    • The number of non-identity elements that are their own inverses

  17. A group has order 8. Could it have a subgroup of order 3?

    • Yes, provided the group is non-abelian
    • Yes, since 3 is less than 8
    • No, since 3 does not divide 8
    • Yes, since every group has subgroups of every order

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

    • 4
    • 5
    • 1
    • 2

  19. Which statement about cyclic and abelian groups is correct?

    • Cyclic groups are never abelian
    • Every cyclic group is abelian, but not every abelian group is cyclic
    • A group is cyclic if and only if it is non-abelian
    • Every abelian group is cyclic

  20. Does Lagrange's theorem show that a group of prime order has only trivial subgroups?

    • Yes, since the only divisors of a prime p are 1 and p, so any subgroup has order 1 or p
    • No, since a group of prime order always has a subgroup of order 2
    • No, since subgroups can have any order
    • Yes, but only when the group is non-abelian

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