8.03a-c Binary operations and the definition of a group Quiz: OCR Further Maths, Unit 5

The 20 questions

1. What property ensures that combining any two elements in set S yields an element in S?

   • Commutativity
   • Associativity
   • Closure
   • Distributivity

2. Which operation on the set of natural numbers N is NOT closed?

   • Exponentiation
   • Subtraction
   • Multiplication
   • Addition

3. What group axiom requires that (a * b) * c = a * (b * c) for all elements?

   • Commutativity
   • Identity
   • Closure
   • Associativity

4. In a group (G, *), what is the result of evaluating e * a?

   • a
   • e^-1
   • a^-1
   • e

5. What is the inverse of the element x in the additive group (R, +)?

   • -x
   • 0
   • x
   • 1/x

6. How many group axioms must a set and binary operation satisfy to form a group?

   • 4
   • 2
   • 5
   • 3

7. What is a group called if its binary operation is commutative for all elements?

   • Finite group
   • Cyclic group
   • Abelian group
   • Symmetric group

8. What is the identity element in the group of non-zero real numbers under multiplication?

   • -1
   • e
   • 0
   • 1

9. What property states that every row in a group Cayley table contains each element exactly once?

   • Latin square
   • Commutativity
   • Associativity
   • Symmetry

10. In any group G, what is the expression for the inverse of the product (ab)^-1?

    • b^-1 a^-1
    • a^-1 b
    • a^-1 b^-1
    • a b^-1

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

    • 0
    • Infinite
    • 1
    • 2

12. In the group (Z_6, +_6), what is the additive inverse of the element 4?

    • 2
    • 4
    • 1
    • 3

13. Which set under standard addition forms a valid mathematical group?

    • Integers
    • Natural numbers
    • Positive integers
    • Odd integers

14. What is the order of the group formed by symmetries of an equilateral triangle?

    • 6
    • 4
    • 8
    • 3

15. What condition must an element a satisfy to be self-inverse in a group?

    • a^2 = 2e
    • a^2 = a
    • a^2 = e
    • a = e

16. Which matrix set forms a non-abelian group under standard matrix multiplication?

    • Diagonal matrices
    • Zero matrices
    • General linear group
    • All square matrices

17. What does the cancellation law ab = ac imply in any group G?

    • a = c
    • b = c
    • a = e
    • b = e

18. How many identity elements can exist within a single group?

    • Exactly 1
    • Zero
    • At least 2
    • Infinitely many

19. What is a group called if it can be generated by powers of a single element?

    • Simple group
    • Cyclic group
    • Symmetric group
    • Abelian group

20. What is the identity element in the group (Z_n, +_n) of integers modulo n?

    • n
    • 0
    • n - 1
    • 1

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