Describe two different isomorphisms between U(20) and U(16)

Question

I'm starting abstract algebra, and I'm currently stuck on this problem: Describe two different isomorphisms between U(20) and U(16).

Not sure to understand how there can be two different isomorphisms between two groups. I thought that if two groups are isomorphic, it means that there exists a bijection that maps the two groups. But in this case, it means that there exist two bijections between U(20) and U(16)?

Could someone clarify/explain how to approach this problem and tell me if I'm right to think that this means that two bijections exist?

Thank you,

Answer 1

We have here several isomorphisms. The first one, arising from CRT, is $$ U(20)\cong U(4) \times U(5), $$ because $\gcd(4,5)=1$. Then we have isomorphisms $$ U(4)\cong C_2,\quad U(5)\cong C_4. $$ Finally we have an isomorphism $$ U(16)\cong C_2 \times C_4. $$ Note that the isomorphisms $U(20)\cong C_2\times C_4$ and $U(16)\cong C_2\times C_4$ need not coincide.