This is part four of a four-part question on Gaussian Integers. I got the other three fairly easily. But now I'm asked to find a "familiar" group and show that $\Bbb Z[i]$ is isomorphic to it. At first, I thought $|x|$ (not absolute value, but length, norm). But that doesn't work. It is a homomorphism but it's not one-to-one as 3+4i and 4+3i would both map to 5. I also tried $\Bbb Z\times\Bbb Z$ but that didn't work either. Can anyone gently steer me in the right direction?
p.s. someone asked for the other three parts, so here they are.
