Order, subgroups and Lagrange's theoremEdexcel A-Level Further Maths: Subtopic test

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

   (a)

   What is the order of the element $10$?

   [1 mark]

   • A$1$
   • B$2$
   • C$5$
   • D$10$

   (b)

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

   [1 mark]

   • A$4$
   • B$6$
   • C$5$
   • D$8$

   (c)

   Find a subgroup of $G$ of order $5$, listing its elements.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $G$ is a finite group of order $24$.

   (a)

   Which of these cannot be the order of a subgroup of $G$?

   [1 mark]

   • A$4$
   • B$8$
   • C$12$
   • D$9$

   (b)

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

   [1 mark]

   • A$16$
   • B$6$
   • C$12$
   • D$8$

   (c)

   An element $g$ of $G$ has order $24$. Explain why $G$ must be cyclic.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The symmetries of an equilateral triangle form a group $T$ of order $6$ under composition. Its elements are the identity $e$, a rotation $r$ through $120^\circ$, the rotation $r^2$ through $240^\circ$, and three reflections $m_1,m_2,m_3$.

   (a)

   State, with justification, the orders of $r$, $r^2$ and $m_1$.

   [3 marks]

   (b)

   Use Lagrange's theorem to explain why $T$ has no subgroup of order $4$, and find a subgroup of order $2$ and a subgroup of order $3$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $G=\{0,1,2,\ldots,11\}$ under addition modulo 12.

   (a)

   Use Lagrange's theorem to list the possible orders of subgroups of $G$, and find a subgroup of each possible order.

   [6 marks]

   (b)

   Find the order of each of the elements $3$, $5$ and $8$. Explain how Lagrange's theorem is consistent with your answers, and deduce whether $G$ has an element of order $5$.

   [6 marks]

   Total for question 4: 12 marks

End of questions