Find an isomorphism from the additive group Z4 04 14 24 34

Find an isomorphism from the additive group Z4 = {[0]4, [1]4, [2]4, [3]4} to the multiplicative group U5 = {[1]5, [2]5, [3]5, [4]5} Z5.

Solution

Since these are cyclic groups, all we need to do is a map a generator for Z4 to a generator of U5.
Note that 1 is a generator of Z4, and 2 is a generator of U5 (2^2 = 4, so 2^4 = 16 = 1 mod 5).

Define f : Z4 U5 by f(1) = 2.
Since we want f to be a homomorphism (at the very least), we must have
f(n) = f(n * 1) = f(1 + ... + 1) = f(1) * ... * f(1) = (f(1))^n = 2^n (mod 5).

In particular:
f(0) = 1 (mod 5)
f(1) = 2 (mod 5)
f(2) = 4 (mod 5)
f(3) = 8 = 3 (mod 5).

Note that f maps distinct elements to distinct elements, and all elements in U5 = {1, 2, 3, 4} are mapped onto by something in f; so f is also a bijection.