Consider:
• $(\mathbb{R}\setminus\{ 0\}, \cdot)$
• $(\mathbb{R}_+, \cdot )$
• $(\mathbb{R} \times \mathbb{Z}_2, +)$, where for all $x, y \in \mathbb{R}$ and all $a, b \in \mathbb{Z}_2$ we define $(x, a) + (y, b) = (x + y, a + b).$
You may assume that these are all groups.
(a) Are the groups $(\mathbb{R}\setminus\{ 0\}, ·)$ and $(\mathbb{R}_+, ·)$ isomorphic? Explain why / why not.
(b) Show that the groups $(\mathbb{R}\setminus 0, ·)$ and $(\mathbb{R} \times \mathbb{Z}_2, +)$ are isomorphic. Hint: Your task is to find (with proof ) a bijection $f : \mathbb{R}\setminus \{0\} \to \mathbb{R}\times \mathbb{Z}_2$ that satisfies $f(x) + f(y) = f(x \cdot y)$ for all $x, y \in \mathbb{R}\setminus \{0\}$.
I've been given these 2 questions for homework but I'm struggling quite a bit with them. Thanks to anyone who is able to help.