How to find quotient groups? I am struggling to understand how do quotient groups work. For example - for the quotient of real numbers (without zero) under multiplication and positive real numbers, the correct quotient should be isomorphic to Z2. Why is that? How can I find what is it isomorphic to? How does in this case the quotient group "divide" the group?

Define
$$\phi:\Bbb R^*\to\{-1\,,\,1\}\le\Bbb C^*\;,\;\;\phi(r):=\begin{cases}\!\!-1&,\;\;r<0\\{}\\\;\,1&,\;\;r>0\end{cases}$$
Check the above is a group homomorphism, that $\;\ker\phi=\Bbb R^*_+\;$ and apply the first isomorphism theorem.
