So I am given two groups and I need to prove that they are not isomorphic. According to the definition of isomorphism, the condition $f(xy) = f(x)f(y)$ should be satisfied. Does it really matter how I define a function though?
For example, my function is $f(x) = x$ and my groups are $(\Bbb Q,+)$ and $(\Bbb Q_{>0}, \cdot)$. When I take arbitrary values from a group and substitute into $f(xy) = f(x)f(y)$ I get a contradiction but I am unsure whether I've really proved that they are not isomorphic.