GroupsAQA A-Level Further Maths: Subtopic test

1. 1
   The set $G=\{1,5,7,11\}$ under multiplication modulo 12.

   (a)

   How many elements of $G$ are their own inverse?

   [1 mark]

   • A$1$
   • B$2$
   • C$4$
   • D$3$

   (b)

   What is the order of the element $7$?

   [1 mark]

   • A$2$
   • B$1$
   • C$3$
   • D$4$

   (c)

   Determine whether $G$ is cyclic, justifying your answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The set $T$ of symmetries of an equilateral triangle forms a group under composition: the identity, rotations through $120^\circ$ and $240^\circ$ about the centre, and reflections in the three lines of symmetry.

   (a)

   What is the order of the group $T$?

   [1 mark]

   • A$3$
   • B$6$
   • C$9$
   • D$12$

   (b)

   What is the order of the rotation through $120^\circ$?

   [1 mark]

   • A$1$
   • B$2$
   • C$6$
   • D$3$

   (c)

   Explain why $T$ is not abelian.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The set $G=\{0,1,2,3,4,5\}$ under addition modulo 6, written $+_6$.

   (a)

   Given that $G$ is closed under $+_6$, show that $(G,+_6)$ satisfies the other three group axioms.

   [3 marks]

   (b)

   Show that $G$ is cyclic, and find all of its proper non-trivial subgroups.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The set $G=\{1,2,4,7,8,11,13,14\}$ is a group under multiplication modulo 15.

   (a)

   (i) Find the order of each element of $G$.
   (ii) Explain why $G$ is not cyclic.

   [6 marks]

   (b)

   (i) Show that $H=\{1,4,11,14\}$ is a subgroup of $G$.
   (ii) Find a cyclic subgroup of $G$ of order $4$ and state a generator.

   [6 marks]

   Total for question 4: 12 marks

End of questions