Isomorphism of groupsAQA A-Level Further Maths: Subtopic test

1. 1
   The set $G=\{1,i,-1,-i\}$ under complex multiplication, and the set $H=\{0,1,2,3\}$ under addition modulo 4. The function $\phi:H\to G$ is defined by $\phi(n)=i^n$.

   (a)

   Which element of $G$ has order $2$?

   [1 mark]

   • A$-1$
   • B$i$
   • C$-i$
   • D$1$

   (b)

   Find $\phi(3)$.

   [1 mark]

   • A$i$
   • B$-1$
   • C$1$
   • D$-i$

   (c)

   Verify that $\phi(2+_43)=\phi(2)\times\phi(3)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The set $G=\{1,3,5,7\}$ under multiplication modulo 8, and the cyclic group $C_4=\{e,a,a^2,a^3\}$ with $a^4=e$.

   (a)

   How many elements of $G$ have order $2$?

   [1 mark]

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

   (b)

   How many elements of $C_4$ have order $2$?

   [1 mark]

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

   (c)

   Deduce that $G$ and $C_4$ are not isomorphic.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The set $A=\{0,1,2,3,4,5\}$ under addition modulo 6, and the set $B=\{1,2,3,4,5,6\}$ under multiplication modulo 7.

   (a)

   Show that $3$ is a generator of $B$.

   [3 marks]

   (b)

   The function $\phi:A\to B$ defined by $\phi(k)=3^k\pmod7$ is an isomorphism. Find $\phi(2)$ and $\phi(3)$, and find the element of $A$ that $\phi$ maps to $5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The group $T$ of symmetries of an equilateral triangle under composition, and the cyclic group $C=\langle c\rangle$ of order $6$, so $c^6=e$.

   (a)

   Give two different reasons why $T$ and $C$ are not isomorphic.

   [6 marks]

   (b)

   A group $H$ has order $6$ and contains an element $h$ of order $6$.
   (i) Show that $H$ is isomorphic to $C$.
   (ii) State whether $H$ is isomorphic to $T$, with a reason.
   (iii) Find the order of $h^4$.

   [6 marks]

   Total for question 4: 12 marks

End of questions