Thread: Isomorphism Question

Show that U(10)-->Z4 where f(3)=2. Show it's either an isomorphism or it isn't.

U(10) = {1,3,7,9} under multiplication and <3> with order 4.

Z4 is 0,1,2,3 under addition mod 4.

I know from a proposition in the book that U(10)-->Z4 is an isomorphism, but I can't quite figure out how f(3)=2 changes that.

Thanks.

you know that 3 is a generator for U(10), good. that means if f:U(10)-->Z4 is given by f(3) = 2, you should be able to calculate the other images of f by taking images of powers of 3:

f(9) = f(3^2) = f(3) + f(3) = ?
f(7) = f(3^3) = f(3) + f(3) + f(3) = ?
f(1) = f(3^4) = f(3) + f(3) + f(3) = ?

is f bijective?