Let $H=\{(a,a) \mid a \in \Bbb R \setminus \{0\} \}$. Show that $(H, \cdot)$ and $(\Bbb R\setminus\{0\}, \cdot)$ are isomorphic.

Question

Let $H=\{(a,a) \mid a \in \Bbb R \setminus \{0\} \}$. Show that $(H, \cdot)$ and $(\Bbb R\setminus\{0\}, \cdot)$ are isomorphic.

I tried to define $\varphi:H \to \Bbb R\setminus \{0\}, (x,y) \mapsto ax+y$. This map is injective and additive, but it seems to fail the surjectivity and $\varphi(e_H) = a+1\ne e_{\ \Bbb R\setminus\{0\}}=1$. Also I don’t think I can define $(x,y) \mapsto ax+y $ since we have no notion of $+$ as an operation.

Also if I define $\varphi:H \to \Bbb R\setminus \{0\}, (x,y) \mapsto xy $, this fails to be injective already so no luck here.

How should I think about this? It seems that there is some underlying idea here that I’m not seeing? There’s probably uncountable many maps that would satisfy this, but somehow I cannot seem to be even figuring out one…

Answer 1

Define $\varphi:H \to \Bbb R\setminus \{0\}, (x,x) \mapsto x$

Answer 2

Hint. Consider: $f:a\in \mathbb{R}\setminus\{0\}\mapsto (a,a)\in H$.