So recently I have been studying group theory and I just really struggle, when it comes to homomorphisms. So, I thought I would tackle some problems and it went ok for a while until I came across this problem:
Let $(\mathbb{C}\text{ \ }\{0\}, * )$ be a group. Show that $f:(\mathbb{C}\text{ \ }\{0\}, * )→(\mathbb{C}\text{ \ }\{0\}, * )$ given by, $z→(\frac{z}{|z|})^2$ is a group homomorphism. Find a subgroup of $(\mathbb{C}\text{ \ }\{0\}, * )$ that is isomorphic to the quotient group $(\mathbb{C}\text{ \ }\{0\}, * )\text{ / }(\mathbb{R}\text{ \ }\{0\}, * )$
Now, proving the group homorphism wasn't particularly hard and $(\mathbb{R}\text{ \ }\{0\}, * )$ is clearly a normal subgroup of $(\mathbb{C}\text{ \ }\{0\}, * )$. I also know that the quotient group $(\mathbb{C}\text{ \ }\{0\}, * )\text { / }\ker(f)$ is isomorphic to the $Im(f)$. But how does that help me find a subgroup of $(\mathbb{C}\text{ \ }\{0\}, * )$ that is isomorphic to $(\mathbb{C}\text{ \ }\{0\}, * )\text{ / }(\mathbb{R}\text{ \ }\{0\}, * )$?
Would I need to find the kernel of f, so all elements $z$ element of $(\mathbb{C}\text{ \ }\{0\}, * )$, for which $(\frac{z}{|z|})^2 = 1$?