Groups and Cayley tables: AQA A‑Level Further Maths questions

AQA A-Level Further Maths past-paper questions on DG.1 Groups and Cayley tables, 2019–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 17 questions on Groups and Cayley tables, as a list

June 2025, Paper 3 Discrete, Q3 (1 mark): The group ( X, *) is an abelian group of order 4 The elements of X are e, a, b and c where e is the identity element of ( X, *) The following is a partially com
June 2025, Paper 3 Discrete, Q10 (5 marks): The set S contains the elements (a, b) where a, b ∈ ℝ except (0, 0) The group G is formed by S under the binary operation ◊ where (a, b) ◊ (c, d) = (a × d + b ×
June 2025, AS Paper 2 Discrete, Q6 (4 marks): The set S is defined as S O x x : 1 2 The binary operation * is defined as a * b = a + b – 2ab where a and b ∈ S It is given that S has exactly one identity el
June 2024, Paper 3 Discrete, Q1 (1 mark): Which one of the following sets forms a group under the given binary operation? Tick () one box. [1 mark] Set Binary Operation {1, 2, 3} Addition modulo 4 {1,
June 2024, Paper 3 Discrete, Q7 (12 marks): (a) By considering associativity, show that the set of integers does not form a group under the binary operation of subtraction. Fully justify your answer. [2 m
June 2024, AS Paper 2 Discrete, Q4 (4 marks): The set S is defined as S = {1, 2, 3, 4} 4 (a) Complete the Cayley Table shown below for S under the binary operation multiplication modulo 5 [2 marks] ×5 1 2 3
June 2024, AS Paper 2 Discrete, Q7 (5 marks): The binary operation ∇ is defined as a ∇ b = a + b + ab where a, b ∈ ℝ 7 (a) Determine if ∇ is commutative on ℝ Fully justify your answer. [2 marks] 7 (b) Prove
June 2023, AS Paper 2 Discrete, Q5 (7 marks): (a) The set S is defined as S = {0, 1, 2, 3, 4, 5} 5 (a) (i) State the identity element of S under the operation multiplication modulo 6 [1 mark] 5 (a) (ii) An
June 2022, Paper 3 Discrete, Q9 (6 marks): The binary operation × acts on the positive integers x and y such that x × y = x + y + 8 (mod k 2 − 16k + 74) where k is a positive integer. 9 (a) (i) Show that
June 2022, AS Paper 2 Discrete, Q6 (5 marks): The set S is given by S = {A, B, C, D} where A = 1 0 0 0 B = 1 0 0 1 C = 0 0 0 1 D = 0 0 0 0 6 (a) Complete the Cayley table for S under matrix multiplication.
November 2021, Paper 3 Discrete, Q5 (11 marks): (a) Describe the conditions necessary for a set of elements, S, under a binary operation * to form a group. [3 marks] 5 (b) In the multiplicative group of integ
November 2021, AS Paper 2 Discrete, Q2 (4 marks): The set S is given by S = {0, 2, 4, 6} 2 (a) Construct a Cayley table, using the grid below, for S under the binary operation addition modulo 8 [3 marks] 2 (b)
November 2021, AS Paper 2 Discrete, Q4 (5 marks): The binary operation * is defined as a * b = ab + 1 where a, b 2 R 4 (a) Prove that * is commutative on R [2 marks] 4 (b) Prove that * is not associative on R [
November 2020, Paper 3 Discrete, Q2 (1 mark): Which of the following statements is true about the operation of matrix multiplication on the set of all 2 − 2 real matrices? Tick (3) one box. [1 mark] Matrix
November 2020, Paper 3 Discrete, Q6 (8 marks): The group (G, ~) has the elements e, r, r2, q, qr and qr2, where r2 = r ~r , qr = q ~r , qr2 = q ~r2 and e is the identity element of G. The elements q and r ha
November 2020, AS Paper 2 Discrete, Q8 (6 marks): The set S is defined as S = fa, b, c, dg Figure 2 shows a Cayley table for S under the commutative binary operation ≤ Figure 2 ≤ a b c d a a a a a b a d b c c a
June 2019, AS Paper 2 Discrete, Q5 (5 marks): (a) Complete the Cayley table in Figure 1 for multiplication modulo 4 [2 marks] Figure 1 ×4 0 1 2 3 0 1 2 3 5 (b) The set S is defined as S = fa, b, c, dg Figur

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