plumiSubgroups, cyclic groups and Lagrange: OCR A‑Level Further Maths questions
OCR A-Level Further Maths past-paper questions on 7.2 Subgroups, cyclic groups and Lagrange, 2018–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.
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All 14 questions on Subgroups, cyclic groups and Lagrange, as a list
- June 2025, Additional Pure Mathematics, Q6 (13 marks): In this question, the set S is {2, 4, 6, 8}, and is the operation of multiplication modulo 10. # 10 (a) The group G consists of the set S, under the operation .
- June 2025, AS Additional Pure Mathematics, Q4 (13 marks): The binary operation * is defined on the set A = {1, 3, 5, 7, 9} by x * y = x + y + 3 (mod 10) for all x, y ! A. (a) (i) Show that * is associative on the eleme
- June 2024, Additional Pure Mathematics, Q8 (15 marks): The group G is cyclic and of order 12. (a) (i) State the possible orders of all the proper subgroups of G. You must justify your [2] answers. (ii) List all the
- June 2024, AS Additional Pure Mathematics, Q5 (14 marks): The set S consists of all 2◊2 matrices having determinant 1 or -1. For instance, the matrices J N J N 3 3 J N 1 1 1 0 K O K O 2 2 2 2 KK OO are elements of S. I
- June 2023, Additional Pure Mathematics, Q9 (11 marks): The set C consists of the set of all complex numbers excluding 1 and - . The operation 5 is 1 a b + defined on the elements of C by a b where ,a b C . 5 = ! ab
- June 2023, AS Additional Pure Mathematics, Q7 (14 marks): The group G, of order 12, consists of the set {1, 2, 4, 5, 8, 10, 13, 16, 17, 19, 20, x} under the operation of multiplication modulo 21. The identity of G is t
- June 2022, Additional Pure Mathematics, Q8 (10 marks): (a) Explain why all groups of even order must contain at least one self-inverse element (that is, an element of order 2). [2] (b) Prove that any group, in which
- June 2022, AS Additional Pure Mathematics, Q7 (13 marks): The diagram below shows an equilateral triangle ABC. The three lines of reflection symmetry of ABC (the lines a, b and c) are shown as broken lines. The point o
- November 2021, Additional Pure Mathematics, Q2 (5 marks): The following Cayley table is for G, a group of order 6. The identity element is e and the group is generated by the elements a and b. a2 2 G e a b ab a b a2 2
- November 2021, AS Additional Pure Mathematics, Q6 (11 marks): The set S consists of the following four complex numbers. 3 i 3 i 1 i 3 1 i 3 + - - - - + ^ h 1 For z1, z S , the binary operation & is defined by z z 1 i 3 z z
- November 2020, Additional Pure Mathematics, Q6 (10 marks): The group G consists of the set {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36} under , the operation # 39 of multiplication modulo 39. (a) List the possible orde
- November 2020, AS Additional Pure Mathematics, Q4 (12 marks): (a) For the set S { , , , , 2 4 6 8 10 12 , }, under the operation #14 of multiplication modulo 14, = complete the Cayley table given in the Printed Answer Book
- June 2019, Additional Pure Mathematics, Q5 (11 marks): The group G consists of a set S together with , the operation of multiplication modulo 80. It is # 80 given that S is the smallest set which contains the elemen
- June 2019, AS Additional Pure Mathematics, Q7 (12 marks): You are given the set S = {1, 5, 7, 11, 13, 17} together with , the operation of multiplication # 18 modulo 18. (a) Complete the Cayley table for (S, ) given in
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