Further Pure 2: GroupsEdexcel A-Level Further Maths: Topic test

1. 1
   The set $G=\{1,2,4,5,7,8\}$ under multiplication modulo 9 is a group.

   (a)

   What is the inverse of $2$ in $G$?

   [1 mark]

   • A$4$
   • B$5$
   • C$7$
   • D$8$

   (b)

   What is the order of the element $4$ in $G$?

   [1 mark]

   • A$2$
   • B$6$
   • C$4$
   • D$3$

   (c)

   Show that $G$ is a cyclic group generated by $2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $G$ is a finite group of order $18$.

   (a)

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

   [1 mark]

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

   (b)

   $G$ contains an element $g$ of order $6$. What is the order of $g^4$?

   [1 mark]

   • A$2$
   • B$4$
   • C$3$
   • D$6$

   (c)

   Explain why $G$ has no subgroup of order $4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The symmetric group $S_6$ of all permutations of $\{1,2,3,4,5,6\}$ has order $720$. Let $x=(1\ 2)(3\ 4\ 5)$ and $y=(1\ 2\ 3\ 4)(5\ 6)$, each written in cycle notation.

   (a)

   Find the order of $x$ and the order of $y$.

   [3 marks]

   (b)

   Write $x^2$ and $x^3$ in cycle notation, state their orders, and explain how the order of the cyclic subgroup $\langle x\rangle$ is consistent with Lagrange's theorem.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   $G=\{1,5,7,11,13,17\}$ is the group of integers coprime to $18$ under multiplication modulo $18$. $H$ is the group of rotations of a regular hexagon about its centre, under composition.

   (a)

   Show that $G$ is cyclic and find the order of every element of $G$.

   [6 marks]

   (b)

   Prove that $G$ is isomorphic to $H$, and find all the subgroups of $G$ other than $\{1\}$ and $G$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   $G=\{1,3,5,7,9,11,13,15\}$ under multiplication modulo $16$ and $H=\{0,1,2,3,4,5,6,7\}$ under addition modulo $8$ are both groups of order $8$.

   (a)

   What is the order of the element $3$ in $G$?

   [1 mark]

   • A$2$
   • B$8$
   • C$6$
   • D$4$

   (b)

   How many elements of order $2$ does $H$ have?

   [1 mark]

   • A$0$
   • B$1$
   • C$2$
   • D$4$

   (c)

   Show that $G$ and $H$ are not isomorphic.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A binary operation $*$ is defined on the set of real numbers by $a*b=a+b-3$.

   (a)

   What is the identity element for $*$?

   [1 mark]

   • A$0$
   • B$-3$
   • C$3$
   • D$6$

   (b)

   What is the inverse of $5$ under $*$?

   [1 mark]

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

   (c)

   Show that $*$ is associative.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   $G=\{1,3,7,9,11,13,17,19\}$ is the group of integers coprime to $20$ under multiplication modulo $20$.

   (a)

   Show that $H=\{1,9,11,19\}$ is a subgroup of $G$.

   [3 marks]

   (b)

   Find the order of the element $3$ in $G$ and explain, using Lagrange's theorem, why every subgroup of $G$ containing $3$ has order $4$ or $8$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   $G=\{1,2,4,7,8,11,13,14\}$ is the group of integers coprime to $15$ under multiplication modulo $15$. It has order $8$.

   (a)

   Find the order of every element of $G$, and deduce that $G$ is not cyclic.

   [6 marks]

   (b)

   Find a cyclic subgroup of order $4$ and a non-cyclic subgroup of order $4$ in $G$, and explain why they are not isomorphic.

   [6 marks]

   Total for question 8: 12 marks

End of questions