Lagrange's theorem and generatorsAQA A-Level Further Maths: Subtopic test

1. 1
   $G$ is a group of order $12$.

   (a)

   Which of the following could be the order of a subgroup of $G$?

   [1 mark]

   • A$5$
   • B$8$
   • C$9$
   • D$6$

   (b)

   Which of the following cannot be the order of an element of $G$?

   [1 mark]

   • A$3$
   • B$8$
   • C$4$
   • D$6$

   (c)

   $G$ contains an element $g$ of order $6$. $K$ is a subgroup of $G$ that contains $g$. Find the possible orders of $K$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $G$ is a group of order $7$.

   (a)

   How many subgroups does $G$ have?

   [1 mark]

   • A$2$
   • B$1$
   • C$3$
   • D$7$

   (b)

   How many elements of $G$ have order $7$?

   [1 mark]

   • A$1$
   • B$7$
   • C$6$
   • D$3$

   (c)

   Explain why $G$ must be cyclic.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   $G=\langle a\rangle$ is a cyclic group of order $18$, with identity $e$, so $a^{18}=e$.

   (a)

   Find the order of each of $a^4$, $a^6$ and $a^9$.

   [3 marks]

   (b)

   (i) Find all the generators of $G$.
   (ii) Find the subgroup of $G$ of order $6$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The set $G=\{1,2,3,\dots,10\}$ is a group under multiplication modulo $11$.

   (a)

   (i) Show that $2$ is a generator of $G$.
   (ii) Hence find all the generators of $G$.

   [6 marks]

   (b)

   (i) Use Lagrange's theorem to list the possible orders of subgroups of $G$.
   (ii) Find the subgroup of order $5$.
   (iii) Find the subgroup of order $2$, and explain why there is only one.

   [6 marks]

   Total for question 4: 12 marks

End of questions