IsomorphismEdexcel A-Level Further Maths: Mind map

Exam questions on Isomorphism

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 $4$.
   Explain why no isomorphism from $G$ to $H$ can map $i$ to $2$.2 marks
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 $4$.
   Give a second reason, based on whether the groups are cyclic, why $K$ and $Z_4$ are not isomorphic.2 marks
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 $\varphi(k)=3^{k}\bmod 7$.
   Find $\varphi(k)$ for each $k\in G$, and hence show that $\varphi$ is a bijection.3 marks