Lesson DG5-DG8

DG5-DG8 Identity, inverses and the group axioms Quiz: AQA Further Maths, Unit 5

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson DG5-DG8, Identity, inverses and the group axioms: 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 is the identity element e for a binary operation * on a set?

    • An element e such that e * a = e for every element a in the set
    • An element e such that e * a = a * e = a for every element a in the set
    • The largest element of the set under the usual ordering
    • An element e such that e * e = e and no other element satisfies this

  2. What is the inverse of an element a, when the set has an identity e?

    • An element b with a * b = a for every element a
    • An element b with a * b = b * a = e
    • An element b with b * b = a
    • An element b, equal to a, for which a * b = b * a only when a = b

  3. Which four properties make up the group axioms?

    • Closure, commutativity, the existence of an identity, and the existence of inverses
    • Closure, associativity, the existence of an identity, and the existence of inverses for every element
    • Associativity, commutativity, a zero element, and the existence of square roots
    • Closure, associativity, and the set being finite

  4. In Z5 under addition modulo 5, what is the identity element?

    • 1
    • 4
    • 0
    • 5

  5. In Z5 under addition modulo 5, what is the inverse of 3?

    • 1
    • 2
    • 3
    • 4

  6. In the set {1, 2, 3, 4} under multiplication modulo 5, what is the inverse of 2?

    • 2
    • 4
    • 1
    • 3

  7. Is {1, 2, 3, 4} under multiplication modulo 5 a group?

    • No, since the set has no identity because 0 is missing
    • No, since multiplication modulo 5 is not associative
    • Yes, since it is closed, associative, has identity 1, and every element has an inverse
    • No, since 2 has no inverse in the set

  8. Are the positive integers under multiplication a group?

    • Yes, since 1 is an identity and every element has an inverse
    • No, since the identity would be 0, which is not positive
    • No, since 2 has no inverse among the positive integers, as 1/2 is not an integer
    • Yes, since multiplication is commutative and associative

  9. Are the integers under addition a group?

    • No, since there is no identity element
    • Yes, with identity 1 and the inverse of a being 1/a
    • No, since addition is not associative on the integers
    • Yes, with identity 0 and the inverse of a being -a

  10. Why is the identity element of a group unique?

    • Because only one element can be paired with itself under the operation
    • Because the identity equals the product of all the elements of the set
    • Because the identity is always the largest element of the set
    • If e and f are both identities, then e = e * f = f, so there can be only one identity

  11. Why is the inverse of an element in a group unique?

    • Because every element has exactly one inverse by the definition of a set
    • Because the inverse is always the largest element of the set
    • If b and c are both inverses of a, associativity gives b = b * e = b * (a * c) = (b * a) * c = e * c = c
    • Because the inverse of a is always a itself

  12. In the Cayley table of a group, where does the identity element appear?

    • Only on the leading diagonal of the table
    • Nowhere, since the identity is not an element that appears in the table
    • Once in every row and once in every column
    • Only in the first row of the table

  13. Is the set {0, 1, 2, 3} under addition modulo 4 a group?

    • Yes, with identity 1 and each element its own inverse
    • No, since 2 has no inverse
    • Yes, with identity 0 and inverses 0, 3, 2, 1 for the elements 0, 1, 2, 3
    • No, since addition modulo 4 is not closed

  14. Why is the set {0, 1, 2, 3} under multiplication modulo 4 not a group?

    • Because 0 has no inverse, and 2 has no inverse since no product 2 × x gives 1 modulo 4
    • Because multiplication modulo 4 is not closed
    • Because the set has no identity element
    • Because multiplication modulo 4 is not associative

  15. What is the identity matrix for 2 × 2 matrices under multiplication?

    • The 2 × 2 zero matrix
    • The matrix with every entry equal to 1
    • The matrix with 1 on the leading diagonal and 2 elsewhere
    • The matrix with 1 on the leading diagonal and 0 elsewhere

  16. Which of the following is the closure axiom for a group?

    • Combining elements in any order gives the same result
    • The result of combining any two elements of the set is also an element of the set
    • The set contains an identity element
    • Every element has an inverse within the set

  17. Is the set {2, 4, 6, 8} under multiplication modulo 10 a group?

    • Yes, it is a group with identity 6
    • No, since the set has no identity element
    • Yes, with identity 2
    • No, since 2 has no inverse in the set

  18. What is the inverse of 4 in Z5 under addition modulo 5?

    • 0
    • 1
    • 2
    • 4

  19. Which statement about the group axioms is correct?

    • A set with associativity and an identity is always a group
    • Commutativity is required for every group
    • A set is a group provided it is closed, even if it has no identity
    • All the group axioms must hold, so a set with an identity but without inverses is not a group

  20. Is the set of all numbers of the form 2^k, where k is an integer, a group under multiplication?

    • No, since the set contains no inverses
    • Yes, with identity 0 and the inverse of 2^k being -2^k
    • No, since products of these numbers are not always of this form
    • Yes, with identity 1 and the inverse of 2^k being 2^(-k)

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