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

The 20 questions

1. Which equation defines the identity element $e$ for all elements $a$ under binary operation $*$?

   • $e * a = e$
   • $e * a = a$
   • $a * a = e$
   • $e * a = 1$

2. For identity element $e$, which equation defines the inverse element $b$ of $a$?

   • $a * b = a$
   • $a * b = e$
   • $a * a = b$
   • $a * b = 1$

3. Which property is NOT one of the four essential group axioms?

   • Closure
   • Commutativity
   • Associativity
   • Identity

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. What is the inverse of $2$ in the group composition of modulo $5$ multiplication?

   • $4$
   • $2$
   • $3$
   • $1$

8. Why do the positive integers under multiplication fail to form a group?

   • Lack of identity
   • Not closed
   • Lack of inverses
   • Not associative

9. What is the identity element of the set of integers under addition?

   • $1$
   • $-1$
   • Does not exist
   • $0$

10. How many identity elements can exist in any single group?

    • Infinitely many
    • Exactly two
    • Zero
    • Exactly one

11. How many inverses does each element have in a group?

    • Zero
    • Infinitely many
    • Exactly one
    • At least two

12. How many times does the identity element appear in each row of a group's Cayley table?

    • Twice
    • In every cell
    • Exactly once
    • Never

13. What is the inverse of $3$ in the group composition of modulo $4$ addition?

    • $2$
    • $0$
    • $1$
    • $3$

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

    • 0 lacks inverse
    • 1 lacks inverse
    • Not closed
    • Not associative

15. Which $2 \times 2$ matrix is the identity element under matrix multiplication?

    • $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$
    • $\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$
    • $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
    • $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

16. Which condition defines the closure axiom for a set $G$ under binary operation $*$?

    • $a * e = a$
    • $a * b \in G$
    • $a * b = e$
    • $a * b = b * a$

17. What is the identity element of the set {2, 4, 6, 8} under multiplication modulo 10?

    • $6$
    • $2$
    • $8$
    • $4$

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

    • 0
    • 1
    • 2
    • 4

19. How many of the four group axioms must hold for a structure to be a group?

    • At least one
    • At least three
    • At least two
    • All four

20. What is the identity element of the group {2^k : k ∈ ℤ} under multiplication?

    • 0
    • 2^k
    • 2
    • 1

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