Isomorphism between groups defined by set builder notation How is it possible to show that for a set $G = \{x + y\sqrt{3}| x, y\in\mathbb{Z}\}$ and a set $H = \{2^i*5^j| i,j\in\mathbb{Z}\}$ that the group $[A, +]$ is isomorphic to $[B, *]$ (let $+$ be standard addition and $*$ be standard multiplication between reals)? In particular I am struggling to find a valid function that holds for the homomorphism property.

Since by definition any number g in G can be represented by (x,y) such that g=x+y*sqrt(3), all you have to show is that the mapping of elements from G to ZXZ is one-one and from there you can use those guaranteed unique x's and y's to go to 2^x*5^y and the isomorphism will be complete. While it would be interesting to find a way to extract the x and y from just the numerical value of the element in G you're already given their existence so the one-one mapping is all you need.
