IsomorphismEdexcel A-Level Further Maths: Subtopic test

1. 1
   Let $G=\{1,i,-1,-i\}$ under multiplication of complex numbers and $H=\{0,1,2,3\}$ under addition modulo 4. Both are groups of order $4$.

   (a)

   Which function $\varphi:G\to H$ is an isomorphism?

   [1 mark]

   • A$\varphi(1)=0,\ \varphi(i)=2,\ \varphi(-1)=1,\ \varphi(-i)=3$
   • B$\varphi(1)=1,\ \varphi(i)=2,\ \varphi(-1)=3,\ \varphi(-i)=0$
   • C$\varphi(1)=0,\ \varphi(i)=1,\ \varphi(-1)=2,\ \varphi(-i)=3$
   • D$\varphi(1)=0,\ \varphi(i)=1,\ \varphi(-1)=3,\ \varphi(-i)=2$

   (b)

   Another isomorphism $\psi:G\to H$ has $\psi(i)=3$. What is $\psi(-1)$?

   [1 mark]

   • A$2$
   • B$1$
   • C$3$
   • D$0$

   (c)

   Explain why no isomorphism from $G$ to $H$ can map $i$ to $2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $K=\{1,3,5,7\}$ under multiplication modulo 8 and $Z_4=\{0,1,2,3\}$ under addition modulo 4. Both are groups of order $4$.

   (a)

   Which statement shows that $K$ and $Z_4$ are not isomorphic?

   [1 mark]

   • A$K$ is abelian but $Z_4$ is not.
   • B$K$ has more elements than $Z_4$.
   • COnly $Z_4$ has an identity element.
   • D$K$ has three elements of order $2$, but $Z_4$ has only one.

   (b)

   The rotation symmetries of a square form a group $R$ of order $4$ that is isomorphic to $Z_4$. How many elements of $R$ have order $4$?

   [1 mark]

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

   (c)

   Give a second reason, based on whether the groups are cyclic, why $K$ and $Z_4$ are not isomorphic.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $G=\{0,1,2,3,4,5\}$ under addition modulo 6 and $H=\{1,2,3,4,5,6\}$ under multiplication modulo 7, both of which are groups. Define $\varphi:G\to H$ by $\varphi(k)=3^{k}\bmod 7$.

   (a)

   Find $\varphi(k)$ for each $k\in G$, and hence show that $\varphi$ is a bijection.

   [3 marks]

   (b)

   Verify that $\varphi(2+3)=\varphi(2)\varphi(3)$, and explain why $\varphi(a+b)=\varphi(a)\varphi(b)$ for all $a,b\in G$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $Z_6=\{0,1,2,3,4,5\}$ under addition modulo 6, and let $S_3$ be the group of all permutations of $\{1,2,3\}$ under composition, where $xy$ means apply $y$ first, then $x$.

   (a)

   Show that $Z_6$ and $S_3$ are not isomorphic, using two different properties of the groups.

   [6 marks]

   (b)

   Let $\omega=\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}$ and let $U=\{1,\omega,\omega^2,\omega^3,\omega^4,\omega^5\}$, the sixth roots of unity, under multiplication. Show that $\theta:Z_6\to U$ given by $\theta(k)=\omega^k$ is an isomorphism.

   [6 marks]

   Total for question 4: 12 marks

End of questions