complex isomorphisms

[math_stresser]

Let G= {1,-1,i,-i} be our well-known group of complex fourth roots of unity under multiplication and let G'=[a]={e,a,a^2,a^3} be a multiplicative cyclic group of order 4. Give two distinct isomorphisms from G to G'.

If we let 1 map to e and i map to a, then we can see that it is:
f(1)=e
f(i)=a
f(-1)=a^2
f(-i)=a^3

I have no idea how else to map this, though.

 

[daon]

You already know 1->e (has to be true for an isomorphism). Now, find the order of each element. An element of order k in G must have an image of order k in G'.

For G, |-1|=2, |i|=4, |-i|=4

For G', |a|=4, |a^2|=2, |a^3|=4

From this you can determine (-1)->\(\displaystyle a^2\) for every isomorphism since they are the only elements of order 2. The only two possible isomorphims are determined by the remaining elements. You know that sending (i)->(a) works. What about sending (i)->(\(\displaystyle a^3\)